Sigma-algebra From Wikipedia, the free encyclopedia Chapter 1

Algebra of sets

The algebra of sets defines the properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evalu- ating expressions, and performing calculations, involving these operations and relations. Any set of sets closed under the set-theoretic operations forms a Boolean algebra with the join operator being union, the meet operator being intersection, and the complement operator being set complement.

1.1 Fundamentals

The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation “less than or equal” is reflexive, antisymmetric and transitive, so is the set relation of “subset”. It is the algebra of the set-theoretic operations of union, intersection and complementation, and the relations of equality and inclusion. For a basic introduction to sets see the article on sets, for a fuller account see naive set theory, and for a full rigorous axiomatic treatment see axiomatic set theory.

1.2 The fundamental laws of set algebra

The binary operations of set union ( ∪ ) and intersection ( ∩ ) satisfy many identities. Several of these identities or “laws” have well established names.

Commutative laws:

• A ∪ B = B ∪ A • A ∩ B = B ∩ A

Associative laws:

• (A ∪ B) ∪ C = A ∪ (B ∪ C) • (A ∩ B) ∩ C = A ∩ (B ∩ C)

Distributive laws:

• A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

The analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking. Like addition and multiplication, the operations of union and intersection are commutative and associative, and inter- section distributes over unions. However, unlike addition and multiplication, union also distributes over intersection.

2 1.3. THE PRINCIPLE OF DUALITY 3

Two additional pairs of laws involve the special sets called the empty set Ø and the universal set U ; together with the complement operator (AC denotes the complement of A). The empty set has no members, and the universal set has all possible members (in a particular context).

Identity laws:

• A ∪ ∅ = A • A ∩ U = A

Complement laws:

• A ∪ AC = U • A ∩ AC = ∅

The identity laws (together with the commutative laws) say that, just like 0 and 1 for addition and multiplication, Ø and U are the identity elements for union and intersection, respectively. Unlike addition and multiplication, union and intersection do not have inverse elements. However the complement laws give the fundamental properties of the somewhat inverse-like unary operation of set complementation. The preceding five pairs of laws—the commutative, associative, distributive, identity and complement laws—encompass all of set algebra, in the sense that every valid proposition in the algebra of sets can be derived from them. Note that if the complement laws are weakened to the rule (AC )C = A , then this is exactly the algebra of proposi- tional linear logic.

1.3 The principle of duality

See also: Duality (order theory)

Each of the identities stated above is one of a pair of identities such that each can be transformed into the other by interchanging ∪ and ∩, and also Ø and U. These are examples of an extremely important and powerful property of set algebra, namely, the principle of duality for sets, which asserts that for any true statement about sets, the dual statement obtained by interchanging unions and intersections, interchanging U and Ø and reversing inclusions is also true. A statement is said to be self-dual if it is equal to its own dual.

1.4 Some additional laws for unions and intersections

The following proposition states six more important laws of set algebra, involving unions and intersections. PROPOSITION 3: For any subsets A and B of a universal set U, the following identities hold:

idempotent laws:

• A ∪ A = A • A ∩ A = A

domination laws:

• A ∪ U = U • A ∩ ∅ = ∅

absorption laws:

• A ∪ (A ∩ B) = A • A ∩ (A ∪ B) = A 4 CHAPTER 1. ALGEBRA OF SETS

As noted above, each of the laws stated in proposition 3 can be derived from the five fundamental pairs of laws stated above. As an illustration, a proof is given below for the idempotent law for union. Proof: The following proof illustrates that the dual of the above proof is the proof of the dual of the idempotent law for union, namely the idempotent law for intersection. Proof: Intersection can be expressed in terms of set difference : A ∩ B = A ∖ (A ∖ B)

1.5 Some additional laws for complements

The following proposition states five more important laws of set algebra, involving complements. PROPOSITION 4: Let A and B be subsets of a universe U, then:

De Morgan’s laws:

• (A ∪ B)C = AC ∩ BC • (A ∩ B)C = AC ∪ BC

double complement or Involution law:

C • (AC ) = A

complement laws for the universal set and the empty set:

• ∅C = U • U C = ∅

Notice that the double complement law is self-dual. The next proposition, which is also self-dual, says that the complement of a set is the only set that satisfies the complement laws. In other words, complementation is characterized by the complement laws. PROPOSITION 5: Let A and B be subsets of a universe U, then:

uniqueness of complements:

• If A ∪ B = U , and A ∩ B = ∅ , then B = AC

1.6 The algebra of inclusion

The following proposition says that inclusion, that is the binary relation of one set being a subset of another, is a partial order. PROPOSITION 6: If A, B and C are sets then the following hold:

reflexivity:

• A ⊆ A

antisymmetry:

• A ⊆ B and B ⊆ A if and only if A = B

transitivity: 1.7. THE ALGEBRA OF RELATIVE COMPLEMENTS 5

• If A ⊆ B and B ⊆ C , then A ⊆ C

The following proposition says that for any set S, the power set of S, ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra. PROPOSITION 7: If A, B and C are subsets of a set S then the following hold:

existence of a least element and a greatest element: • ∅ ⊆ A ⊆ S

existence of joins: • A ⊆ A ∪ B • If A ⊆ C and B ⊆ C , then A ∪ B ⊆ C

existence of meets: • A ∩ B ⊆ A • If C ⊆ A and C ⊆ B , then C ⊆ A ∩ B

The following proposition says that the statement A ⊆ B is equivalent to various other statements involving unions, intersections and complements. PROPOSITION 8: For any two sets A and B, the following are equivalent:

• A ⊆ B • A ∩ B = A • A ∪ B = B • A ∖ B = ∅ • BC ⊆ AC

The above proposition shows that the relation of set inclusion can be characterized by either of the operations of set union or set intersection, which means that the notion of set inclusion is axiomatically superfluous.

1.7 The algebra of relative complements

The following proposition lists several identities concerning relative complements and set-theoretic differences. PROPOSITION 9: For any universe U and subsets A, B, and C of U, the following identities hold:

• C \ (A ∩ B) = (C \ A) ∪ (C \ B) • C \ (A ∪ B) = (C \ A) ∩ (C \ B) • C \ (B \ A) = (A ∩ C) ∪ (C \ B) • (B \ A) ∩ C = (B ∩ C) \ A = B ∩ (C \ A) • (B \ A) ∪ C = (B ∪ C) \ (A \ C) • A \ A = ∅ • ∅ \ A = ∅ • A \ ∅ = A • B \ A = AC ∩ B • (B \ A)C = A ∪ BC • U \ A = AC • A \ U = ∅ 6 CHAPTER 1. ALGEBRA OF SETS

1.8 See also

• σ-algebra is an algebra of sets, completed to include countably infinite operations.

• Axiomatic set theory • Field of sets

• Naive set theory

• Set (mathematics)

1.9 References

• Stoll, Robert R.; Set Theory and Logic, Mineola, N.Y.: Dover Publications (1979) ISBN 0-486-63829-4. “The Algebra of Sets”, pp 16—23 • Courant, Richard, Herbert Robbins, Ian Stewart, What is mathematics?: An Elementary Approach to Ideas and Methods, Oxford University Press US, 1996. ISBN 978-0-19-510519-3. “SUPPLEMENT TO CHAPTER II THE ALGEBRA OF SETS”

1.10 External links

• Operations on Sets at ProvenMath Chapter 2

Join (sigma algebra)

In mathematics, the join of two sigma algebras over the same set X is the coarsest sigma algebra containing both.[1]

2.1 References

[1] Peter Walters, An Introduction to Ergodic Theory (1982) Springer Verlag, ISBN 0-387-90599-5

7 Chapter 3

Measurable function

A function is Lebesgue measurable if and only if the preimage of each of the sets [a, ∞] is a Lebesgue measurable set.

In mathematics, particularly in measure theory, measurable functions are structure-preserving functions between measurable spaces; as such, they form a natural context for the theory of integration. Specifically, a function between measurable spaces is said to be measurable if the preimage of each measurable set is measurable, analogous to the situation of continuous functions between topological spaces. In probability theory, the sigma algebra often represents the set of available information, and a function (in this context a random variable) is measurable if and only if it represents an outcome that is knowable based on the available information. In contrast, functions that are not Lebesgue measurable are generally considered pathological, at least in the field of analysis.

3.1 Formal definition

Let (X, Σ) and (Y, Τ) be measurable spaces, meaning that X and Y are sets equipped with respective sigma algebras Σ and Τ. A function f: X → Y is said to be measurable if the preimage of E under f is in Σ for every E ∈ Τ; i.e.

f −1(E) := {x ∈ X| f(x) ∈ E} ∈ Σ, ∀E ∈ T.

The notion of measurability depends on the sigma algebras Σ and Τ. To emphasize this dependency, if f: X → Y is a measurable function, we will write

f :(X, Σ) → (Y,T )

8 3.2. CAVEAT 9

3.2 Caveat

This definition can be deceptively simple, however, as special care must be taken regarding the σ-algebras involved. In particular, when a function f: R → R is said to be Lebesgue measurable, what is actually meant is that f :(R, L) → (R, B) is a measurable function—that is, the domain and range represent different σ-algebras on the same underlying set. Here, L is the sigma algebra of Lebesgue measurable sets, and B is the Borel algebra on R, the smallest σ- algebra containing all the open sets. As a result, the composition of Lebesgue-measurable functions need not be Lebesgue-measurable. By convention a topological space is assumed to be equipped with the Borel algebra unless otherwise specified. Most commonly this space will be the real or complex numbers. For instance, a real-valued measurable function is a function for which the preimage of each Borel set is measurable. A complex-valued measurable function is defined analogously. In practice, some authors use measurable functions to refer only to real-valued measurable functions with respect to the Borel algebra.[1] If the values of the function lie in an infinite-dimensional vector space instead of R or C, usually other definitions of measurability are used, such as weak measurability and Bochner measurability.

3.3 Special measurable functions

• If (X, Σ) and (Y, Τ) are Borel spaces, a measurable function f:(X, Σ) → (Y, Τ) is also called a Borel function. Continuous functions are Borel functions but not all Borel functions are continuous. However, a measurable function is nearly a continuous function; see Luzin’s theorem. If a Borel function happens to be a section of some map Y →π X , it is called a Borel section.

• A Lebesgue measurable function is a measurable function f :(R, L) → (C, BC) , where L is the sigma algebra of Lebesgue measurable sets, and BC is the Borel algebra on the complex numbers C. Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. In the case f : X → R , f is Lebesgue measurable iff {f > α} = {x ∈ X : f(x) > α} is measurable for all real α . This is also equivalent to any of {f ≥ α}, {f < α}, {f ≤ α} being measurable for all α . Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable.[2] A function f : X → C is measurable iff the real and imaginary parts are measurable.

• Random variables are by definition measurable functions defined on sample spaces.

3.4 Properties of measurable functions

• The sum and product of two complex-valued measurable functions are measurable.[3] So is the quotient, so long as there is no division by zero.[1]

• The composition of measurable functions is measurable; i.e., if f:(X,Σ1) → (Y,Σ2) and g:(Y,Σ2) → (Z, [1] Σ3) are measurable functions, then so is g(f(⋅)): (X,Σ1) → (Z,Σ3). But see the caveat regarding Lebesgue- measurable functions in the introduction.

• The (pointwise) supremum, infimum, limit superior, and limit inferior of a sequence (viz., countably many) of real-valued measurable functions are all measurable as well.[1][4]

• The pointwise limit of a sequence of measurable functions is measurable (if the codomain in endowed with the Borel algebra); note that the corresponding statement for continuous functions requires stronger conditions than pointwise convergence, such as uniform convergence. (This is correct when the counter domain of the elements of the sequence is a metric space. It is false in general; see pages 125 and 126 of.[5])

3.5 Non-measurable functions

Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to find non- measurable functions. 10 CHAPTER 3. MEASURABLE FUNCTION

• So long as there are non-measurable sets in a measure space, there are non-measurable functions from that space. If (X, Σ) is some measurable space and A ⊂ X is a non-measurable set, i.e. if A ∉ Σ, then the indicator function 1A:(X, Σ) → R is non-measurable (where R is equipped with the Borel algebra as usual), since the preimage of the measurable set {1} is the non-measurable set A. Here 1A is given by { 1 if x ∈ A 1A(x) = 0 otherwise

• Any non-constant function can be made non-measurable by equipping the domain and range with appropriate σ-algebras. If f: X → R is an arbitrary non-constant, real-valued function, then f is non-measurable if X is equipped with the indiscrete algebra Σ = {∅, X}, since the preimage of any point in the range is some proper, nonempty subset of X, and therefore does not lie in Σ.

3.6 See also

• Vector spaces of measurable functions: the Lp spaces

• Measure-preserving dynamical system

3.7 Notes

[1] Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6.

[2] Carothers, N. L. (2000). Real Analysis. Cambridge University Press. ISBN 0-521-49756-6.

[3] Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0-471-31716-0.

[4] Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0-02-404151-3.

[5] Dudley, R. M. (2002). Real Analysis and Probability (2 ed.). Cambridge University Press. ISBN 0-521-00754-2.

3.8 External links

• Measurable function at Encyclopedia of Mathematics

• Borel function at Encyclopedia of Mathematics Chapter 4

Pi system

In mathematics, a π-system (or pi-system) on a set Ω is a collection P of certain subsets of Ω, such that

• P is non-empty.

• A ∩ B ∈ P whenever A and B are in P.

That is, P is a non-empty family of subsets of Ω that is closed under finite intersections. The importance of π-systems arise from the fact that if two probability measures agree on a π-system, then they agree on the σ-algebra generated by that π-system. Moreover, if other properties, such as equality of integrals, hold for the π-system, then they hold for the generated σ-algebra as well. This is the case whenever the collection of subsets for which the property holds is a λ-system. π-systems are also useful for checking independence of random variables. This is desirable because in practice, π-systems are often simpler to work with than σ-algebras. For example, it may be awkward to work with σ-algebras generated by infinitely∪ many sets σ(E1,E2,...) . So instead we may examine the union of all σ-algebras generated by finitely many sets n σ(E1,...,En) . This forms a π-system that generates the desired σ-algebra. Another example is the collection of all interval subsets of the real line, along with the empty set, which is a π-system that generates the very important Borel σ-algebra of subsets of the real line.

4.1 Examples

• ∀a, b ∈ R , the intervals (−∞, a] form a π-system, and the intervals (a, b] form a π-system, if the empty set is also included.

• The topology (collection of open subsets) of any topological space is a π-system.

• For any collection Σ of subsets of Ω, there exists a π-system IΣ which is the unique smallest π-system of Ω to contain every element of Σ, and is called the π-system generated by Σ. { } • → R I −1 −∞ ∈ R For any measurable function f :Ω , the set f{= f (( , x]) : x }defines a π-system, and is called the π-system generated by f. (Alternatively, f −1 ((a, b]) : a, b ∈ R, a < b ∪ {∅} defines a π-system generated by f .)

• If P1 and P2 are π-systems for Ω1 and Ω2, respectively, then {A1 × A2 : A1 ∈ P1,A2 ∈ P2} is a π-system for the product space Ω1×Ω2.

• Any σ-algebra is a π-system.

4.2 Relationship to λ-Systems

A λ-system on Ω is a set D of subsets of Ω, satisfying

11 12 CHAPTER 4. PI SYSTEM

• Ω ∈ D ,

• if A ∈ D then Ac ∈ D ,

• ∪∞ ∈ if A1,A2,A3,... is a sequence of disjoint subsets in D then n=1An D .

Whilst it is true that any σ-algebra satisfies the properties of being both a π-system and a λ-system, it is not true that any π-system is a λ-system, and moreover it is not true that any π-system is a σ-algebra. However, a useful classification is that any set system which is both a λ-system and a π-system is a σ-algebra. This is used as a step in proving the π-λ theorem.

4.2.1 The π-λ Theorem

Let D be a λ-system, and let I ⊆ D be a π-system contained in D . The π-λ Theorem[1] states that the σ-algebra σ(I) generated by I is contained in D : σ(I) ⊂ D . The π-λ theorem can be used to prove many elementary measure theoretic results. For instance, it is used in proving the uniqueness claim of the Carathéodory extension theorem for σ-finite measures.[2] The π-λ theorem is closely related to the monotone class theorem, which provides a similar relationship between monotone classes and algebras, and can be used to derive many of the same results. Since π-systems are simpler classes than algebras, it can be easier to identify the sets that are in them while, on the other hand, checking whether the property under consideration determines a λ-system is often relatively easy. Despite the difference between the two theorems, the π-λ theorem is sometimes referred to as the monotone class theorem.[1]

Example

Let μ₁ , μ2 : F → R be two measures on the σ-algebra F, and suppose that F = σ(I) is generated by a π-system I. If

1. μ1(A) = μ2(A), ∀ A ∈ I, and

2. μ1(Ω) = μ2(Ω) < ∞,

then μ₁ = μ2. This is the uniqueness statement of the Carathéodory extension theorem for finite measures. If this result does not seem very remarkable, consider the fact that it usually is very difficult or even impossible to fully describe every set in the σ-algebra, and so the problem of equating measures would be completely hopeless without such a tool. Idea of Proof[2] Define the collection of sets

D = {A ∈ σ(I): µ1(A) = µ2(A)} .

By the first assumption, μ1 and μ2 agree on I and thus I D. By the second assumption, Ω ∈ D, and it can further be shown that D is a λ-system. It follows from the π-λ theorem that σ(I) D σ(I), and so D = σ(I). That is to say, the measures agree on σ(I).

4.3 π-Systems in Probability

π-systems are more commonly used in the study of probability theory than in the general field of measure theory. This is primarily due to probabilistic notions such as independence, though it may also be a consequence of the fact that the π-λ theorem was proven by the probabilist Eugene Dynkin. Standard measure theory texts typically prove the same results via monotone classes, rather than π-systems. 4.3. Π-SYSTEMS IN PROBABILITY 13

4.3.1 Equality in Distribution

The π-λ theorem motivates the common definition of the probability distribution of a random variable X : (Ω, F, P) → R in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as

FX (a) = P [X ≤ a] , a ∈ R whereas the seemingly more general law of the variable is the probability measure

[ ] −1 LX (B) = P X (B) ,B ∈ B(R) where B(R) is the Borel σ-algebra. We say that the random variables X : (Ω, F, P) , and Y :(Ω˜, F˜, P˜) → R (on D two possibly different probability spaces) are equal in distribution (or law), X=Y , if they have the same cumulative distribution functions, FX = FY. The motivation for the definition stems from the observation that if FX = FY, then that is exactly to say that LX and LY agree on the π-system {(−∞, a]: a ∈ R} which generates B(R) , and so by the example above: LX = LY . A similar result holds for the joint distribution of a random vector. For example, suppose X and Y are two random variables defined on the same probability space (Ω, F, P) , with respectively generated π-systems IX and IY . The joint cumulative distribution function of (X,Y) is

[ ] −1 −1 FX,Y (a, b) = P [X ≤ a, Y ≤ b] = P X ((−∞, a]) ∩ Y ((−∞, b]) , a, b ∈ R

−1 −1 However, A = X ((−∞, a]) ∈ IX and B = Y ((−∞, b]) ∈ IY . Since

IX,Y = {A ∩ B : A ∈ IX ,B ∈ IY }

is a π-system generated by the random pair (X,Y), the π-λ theorem is used to show that the joint cumulative distribu- tion function suffices to determine the joint law of (X,Y). In other words, (X,Y) and (W,Z) have the same distribution if and only if they have the same joint cumulative distribution function.

In the theory of stochastic processes, two processes (Xt)t∈T , (Yt)t∈T are known to be equal in distribution if and only if they agree on all finite-dimensional distributions. i.e. for all t1, . . . , tn ∈ T, n ∈ N .

D (Xt1 ,...,Xtn )=(Yt1 ,...,Ytn )

The proof of this is another application of the π-λ theorem.[3]

4.3.2 Independent Random Variables

The theory of π-system plays an important role in the probabilistic notion of independence. If X and Y are two random variables defined on the same probability space (Ω, F, P) then the random variables are independent if and only if their π-systems IX , IY satisfy

P [A ∩ B] = P [A] P [B] , ∀A ∈ IX ,B ∈ IY ,

which is to say that IX , IY are independent. This actually is a special case of the use of π-systems for determining the distribution of (X,Y). 14 CHAPTER 4. PI SYSTEM

Example

Let Z = (Z1,Z2) , where Z1,Z2 ∼ N (0, 1) are iid standard normal random variables. Define the radius and argument (arctan) variables

√ 2 2 −1 R = Z1 + Z2 , Θ = tan (Z2/Z1)

Then R and Θ are independent random variables.

To prove this, it is sufficient to show that the π-systems IR, IΘ are independent: i.e.

P[R ≤ ρ, Θ ≤ θ] = P[R ≤ ρ]P[Θ ≤ θ] ∀ρ ∈ [0, ∞), θ ∈ [0, 2π].

Confirming that this is the case is an exercise in changing variables. Fix ρ ∈ [0, ∞), θ ∈ [0, 2π] , then the probability can be expressed as an integral of the probability density function of Z .

∫ ( ) P ≤ ≤ 1 −1 2 2 [R ρ, Θ θ] = exp (z1 + z2 ) dz1dz2 ≤ ≤ 2π 2 ∫R ∫ρ, Θ θ θ ρ 1 − r2 = e 2 rdrdθ˜ 0 0 2π (∫ ) (∫ ) θ ρ 1 − r2 = dθ˜ e 2 rdr 0 2π 0 = P[Θ ≤ θ]P[R ≤ ρ].

4.4 See also

• λ-systems

• σ-algebra

• Monotone class theorem • Probability distribution

• Independence

4.5 Notes

[1] Kallenberg, Foundations Of Modern Probability, p.2

[2] Durrett, Probability Theory and Examples, p.404

[3] Kallenberg, Foundations Of Modern probability, p. 48

4.6 References

• Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN 0-387-22833-0. • David Williams (1991). Probability with Martingales. Cambridge University Press. ISBN 0-521-40605-6. Chapter 5

Sample space

In probability theory, the sample space[nb 1] of an experiment or random trial is the set of all possible outcomes or results of that experiment.[4] A sample space is usually denoted using set notation, and the possible outcomes are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}. For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).[5] A well-defined sample space is one of three basic elements in a probabilistic model (a probability space); the other two are a well-defined set of possible events (a sigma-algebra) and a probability assigned to each event (a probability measure function).

5.1 Multiple sample spaces

For many experiments, there may be more than one plausible sample space available, depending on what result is of interest to the experimenter. For example, when drawing a card from a standard deck of fifty-two playing cards, one possibility for the sample space could be the various ranks (Ace through King), while another could be the suits (clubs, diamonds, hearts, or spades).[4][6] A more complete description of outcomes, however, could specify both the denomination and the suit, and a sample space describing each individual card can be constructed as the Cartesian product of the two sample spaces noted above (this space would contain fifty-two equally likely outcomes). Still other sample spaces are possible, such as {right-side up, up-side down} if some cards have been flipped when shuffling.

5.2 Equally likely outcomes

Main article: Equally likely outcomes

Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be equally likely.[7] However, there are experiments that are not easily described by a sample space of equally likely outcomes— for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be equally likely. Though most random phenomena do not have equally likely outcomes, it can be helpful to define a sample space in such a way that outcomes are at least approximately equally likely, since this condition significantly simplifies the computation of probabilities for events within the sample space. If each individual outcome occurs with the same probability, then the probability of any event becomes simply:[8]:346–347

event in outcomes of number P (event) = space sample in outcomes of number

15 16 CHAPTER 5. SAMPLE SPACE

Flipping a coin leads to a sample space composed of two outcomes that are almost equally likely.

5.2.1 Simple random sample

Main article: Simple random sample

In statistics, inferences are made about characteristics of a population by studying a sample of that population’s indi- viduals. In order to arrive at a sample that presents an unbiased estimate of the true characteristics of the population, 5.3. INFINITELY LARGE SAMPLE SPACES 17

Up or down? Flipping a brass tack leads to a sample space composed of two outcomes that are not equally likely. statisticians often seek to study a simple random sample— that is, a sample in which every individual in the popula- tion is equally likely to be included.[8]:274–275 The result of this is that every possible combination of individuals who could be chosen for the sample is also equally likely (that is, the space of simple random samples of a given size from a given population is composed of equally likely outcomes).

5.3 Infinitely large sample spaces

In an elementary approach to probability, any subset of the sample space is usually called an event. However, this gives rise to problems when the sample space is infinite, so that a more precise definition of an event is necessary. Under this definition only measurable subsets of the sample space, constituting a σ-algebra over the sample space itself, are considered events. However, this has essentially only theoretical significance, since in general the σ-algebra can always be defined to include all subsets of interest in applications.

5.4 See also

• Probability space • Space (mathematics) • Set (mathematics) 18 CHAPTER 5. SAMPLE SPACE

• Event (probability theory)

• σ-algebra

5.5 Notes

[1] also called sample description space[1] or event space[2] or possibility space[3]

5.6 References

[1] Stark, Henry; Woods, John W. (2002). Probability and Random Processes with Applications to Signal Processing (3rd ed.). Pearson. p. 7. ISBN 9788177583564.

[2] “OECD Glossary of Statistical Terms - Sample space Definition”. May 26, 2002. Retrieved February 27, 2015.

[3] Forbes, Catherine; Evans, Merran; Hastings, Nicholas; Peacock, Brian (2011). Statistical Distributions (4th ed.). Wiley. p. 3. ISBN 9780470390634.

[4] Albert, Jim (21 January 1998). “Listing All Possible Outcomes (The Sample Space)". Bowling Green State University. Retrieved June 25, 2013.

[5] Larsen, R. J.; Marx, M. L. (2001). An Introduction to Mathematical Statistics and Its Applications (Third ed.). Upper Saddle River, NJ: Prentice Hall. p. 22. ISBN 9780139223037.

[6] Jones, James (1996). “Stats: Introduction to Probability - Sample Spaces”. Richland Community College. Retrieved November 30, 2013.

[7] Foerster, Paul A. (2006). Algebra and Trigonometry: Functions and Applications, Teacher’s Edition (Classics ed.). Prentice Hall. p. 633. ISBN 0-13-165711-9.

[8] Yates, Daniel S.; Moore, David S; Starnes, Daren S. (2003). The Practice of Statistics (2nd ed.). New York: Freeman. ISBN 978-0-7167-4773-4. Chapter 6

Separable sigma algebra

In mathematics, σ-algebras are usually studied in the context of measure theory.A separable σ-algebra (or sepa- rable σ-field) is a σ-algebra F which is a separable space when considered as a metric space with metric ρ(A, B) = µ(A△B) for A, B ∈ F and a given measure µ (and with △ being the symmetric difference operator). Note that any σ-algebra generated by a countable collection of sets is separable, but the converse need not hold. For example, the Lebesgue σ-algebra is separable (since every Lebesgue measurable set is equivalent to some Borel set) but not countably generated (since its cardinality is higher than continuum). A separable measure space has a natural pseudometric that renders it separable as a pseudometric space. The distance between two sets is defined as the measure of the symmetric difference of the two sets. Note that the symmetric difference of two distinct sets can have measure zero; hence the pseudometric as defined above need not to be a true metric. However, if sets whose symmetric difference has measure zero are identified into a single equivalence class, the resulting quotient set can be properly metrized by the induced metric. If the measure space is separable, it can be shown that the corresponding metric space is, too.

19 Chapter 7

Sigma additivity

In mathematics, additivity and sigma additivity (also called countable additivity) of a function defined on subsets of a given set are abstractions of the intuitive properties of size (length, area, volume) of a set.

7.1 Additive (or finitely additive) set functions

Let µ be a function defined on an algebra of sets A with values in [−∞, +∞] (see the extended real number line). The function µ is called additive, or finitely additive, if, whenever A and B are disjoint sets in A , one has

µ(A ∪ B) = µ(A) + µ(B). (A consequence of this is that an additive function cannot take both −∞ and +∞ as values, for the expression ∞ − ∞ is undefined.) One can prove by mathematical induction that an additive function satisfies

( ) ∪N ∑N µ An = µ(An) n=1 n=1 for any A1,A2,...,AN disjoint sets in A .

7.2 σ-additive set functions

Suppose that A is a σ-algebra. If for any sequence A1,A2,...,An,... of disjoint sets in A , one has

∪∞ ∑∞ µ ( n=1 An) = n=1 µ(An) ,

we say that μ is countably additive or σ-additive. Any σ-additive function is additive but not vice versa, as shown below.

7.3 τ-additive set functions

Suppose that in addition to a sigma algebra A , we have a topology τ. If for any directed family of measurable open sets G ⊆ A ∩τ, ∪ G µ ( ) = supG∈G µ(G) ,

we say that μ is τ-additive. In particular, if μ is inner regular then it is τ-additive.[1]

20 7.4. PROPERTIES 21

7.4 Properties

7.4.1 Basic properties

Useful properties of an additive function μ include the following:

1. Either μ(∅) = 0, or μ assigns ∞ to all sets in its domain, or μ assigns −∞ to all sets in its domain. 2. If μ is non-negative and A ⊆ B, then μ(A) ≤ μ(B). 3. If A ⊆ B and μ(B) − μ(A) is defined, then μ(B \ A) = μ(B) − μ(A). 4. Given A and B, μ(A ∪ B) + μ(A ∩ B) = μ(A) + μ(B).

7.5 Examples

An example of a σ-additive function is the function μ defined over the power set of the real numbers, such that

{ 1 if 0 ∈ A µ(A) = 0 if 0∈ / A.

If A1,A2,...,An,... is a sequence of disjoint sets of real numbers, then either none of the sets contains 0, or precisely one of them does. In either case, the equality

( ) ∪∞ ∑∞ µ An = µ(An) n=1 n=1 holds. See measure and signed measure for more examples of σ-additive functions.

7.5.1 An additive function which is not σ-additive

An example of an additive function which is not σ-additive is obtained by considering μ, defined over the Lebesgue sets of the real numbers by the formula

1 µ(A) = lim · λ (A ∩ (0, k)) , k→∞ k where λ denotes the Lebesgue measure and lim the Banach limit. One can check that this function is additive by using the linearity of the limit. That this function is not σ-additive follows by considering the sequence of disjoint sets

An = [n, n + 1) for n=0, 1, 2, ... The union of these sets is the positive reals, and μ applied to the union is then one, while μ applied to any of the individual sets is zero, so the sum of μ(An) is also zero, which proves the counterexample.

7.6 Generalizations

One may define additive functions with values in any additive monoid (for example any group or more commonly a vector space). For sigma-additivity, one needs in addition that the concept of limit of a sequence be defined on that set. For example, spectral measures are sigma-additive functions with values in a Banach algebra. Another example, also from quantum mechanics, is the positive operator-valued measure. 22 CHAPTER 7. SIGMA ADDITIVITY

7.7 See also

• signed measure

• measure (mathematics) • additive map

• subadditive function

• σ-finite measure • Hahn–Kolmogorov theorem

This article incorporates material from additive on PlanetMath, which is licensed under the Creative Commons Attribution/Share- Alike License.

7.8 References

[1] D.H. Fremlin Measure Theory, Volume 4, Torres Fremlin, 2003. Chapter 8

Sigma-algebra

"Σ-algebra” redirects here. For an algebraic structure admitting a given signature Σ of operations, see Universal al- gebra.

In mathematical analysis and in probability theory, a σ-algebra (also sigma-algebra, σ-field, sigma-field) on a set X is a collection Σ of subsets of X that is closed under countable-fold set operations (complement, union of countably many sets and intersection of countably many sets). By contrast, an algebra is only required to be closed under finitely many set operations. That is, a σ-algebra is an algebra of sets, completed to include countably infinite operations. The pair (X, Σ) is also a field of sets, called a measurable space. The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation. In statistics, (sub) σ-algebras are needed for a formal mathematical definition of sufficient statistic,[1] particularly when the statistic is a function or a random process and the notion of conditional density is not applicable. If X = {a, b, c, d}, one possible σ-algebra on X is Σ = { ∅, {a, b}, {c, d}, {a, b, c, d} }, where ∅ is the empty set. However, a finite algebra is always a σ-algebra.

If {A1, A2, A3, …} is a countable partition of X then the collection of all unions of sets in the partition (including the empty set) is a σ-algebra. A more useful example is the set of subsets of the real line formed by starting with all open intervals and adding in all countable unions, countable intersections, and relative complements and continuing this process (by transfinite iteration through all countable ordinals) until the relevant closure properties are achieved (a construction known as the Borel hierarchy).

8.1 Motivation

There are at least three key motivators for σ-algebras: defining measures, manipulating limits of sets, and managing partial information characterized by sets.

8.1.1 Measure

A measure on X is a function that assigns a non-negative real number to subsets of X; this can be thought of as making precise a notion of “size” or “volume” for sets. We want the size of the union of disjoint sets to be the sum of their individual sizes, even for an infinite sequence of disjoint sets. One would like to assign a size to every subset of X, but in many natural settings, this is not possible. For example the axiom of choice implies that when the size under consideration is the ordinary notion of length for subsets of the real line, then there exist sets for which no size exists, for example, the Vitali sets. For this reason, one considers instead a smaller collection of privileged subsets of X. These subsets will be called the measurable sets. They are closed under

23 24 CHAPTER 8. SIGMA-ALGEBRA operations that one would expect for measurable sets, that is, the complement of a measurable set is a measurable set and the countable union of measurable sets is a measurable set. Non-empty collections of sets with these properties are called σ-algebras.

8.1.2 Limits of sets

Many uses of measure, such as the probability concept of almost sure convergence, involve limits of sequences of sets. For this, closure under countable unions and intersections is paramount. Set limits are defined as follows on σ-algebras.

• The limit supremum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

∩∞ ∪∞ lim sup An = Am. n→∞ n=1 m=n

• The limit infimum of a sequence A1, A2, A3, ..., each of which is a subset of X, is

∪∞ ∩∞ lim inf An = Am. n→∞ n=1 m=n

• If, in fact,

lim inf A = lim sup A →∞ n n n n→∞

then the limn→∞ An exists as that common set.

8.1.3 Sub σ-algebras

In much of probability, especially when conditional expectation is involved, one is concerned with sets that represent only part of all the possible information that can be observed. This partial information can be characterized with a smaller σ-algebra which is a subset of the principal σ-algebra; it consists of the collection of subsets relevant only to and determined only by the partial information. A simple example suffices to illustrate this idea. Imagine you are playing a game that involves flipping a coin repeatedly and observing whether it comes up Heads (H) or Tails (T). Since you and your opponent are each infinitely wealthy, there is no limit to how long the game can last. This means the sample space Ω must consist of all possible infinite sequences of H or T:

∞ Ω = {H,T } = {(x1, x2, x3,... ): xi ∈ {H,T }, i ≥ 1} However, after n flips of the coin, you may want to determine or revise your betting strategy in advance of the next flip. The observed information at that point can be described in terms of the 2n possibilities for the first n flips. Formally, since you need to use subsets of Ω, this is codified as the σ-algebra

∞ n Gn = {A × {H,T } : A ⊂ {H,T } } Observe that then

G1 ⊂ G2 ⊂ G3 ⊂ · · · ⊂ G∞ where G∞ is the smallest σ-algebra containing all the others. 8.2. DEFINITION AND PROPERTIES 25

8.2 Definition and properties

8.2.1 Definition

Let X be some set, and let 2X represent its power set. Then a subset Σ ⊂ 2X is called a σ-algebra if it satisfies the following three properties:[2]

1. X is in Σ, and X is considered to be the universal set in the following context.

2. Σ is closed under complementation: If A is in Σ, then so is its complement, X\A.

3. Σ is closed under countable unions: If A1, A2, A3, ... are in Σ, then so is A = A1 ∪ A2 ∪ A3 ∪ … .

From these properties, it follows that the σ-algebra is also closed under countable intersections (by applying De Morgan’s laws). It also follows that the empty set ∅ is in Σ, since by (1) X is in Σ and (2) asserts that its complement, the empty set, is also in Σ. Moreover, by (3) it follows as well that {X, ∅} is the smallest possible σ-algebra. Elements of the σ-algebra are called measurable sets. An ordered pair (X, Σ), where X is a set and Σ is a σ-algebra over X, is called a measurable space. A function between two measurable spaces is called a measurable function if the preimage of every measurable set is measurable. The collection of measurable spaces forms a category, with the measurable functions as morphisms. Measures are defined as certain types of functions from a σ-algebra to [0, ∞]. A σ-algebra is both a π-system and a Dynkin system (λ-system). The converse is true as well, by Dynkin’s theorem (below).

8.2.2 Dynkin’s π-λ theorem

This theorem (or the related monotone class theorem) is an essential tool for proving many results about properties of specific σ-algebras. It capitalizes on the nature of two simpler classes of sets, namely the following.

A π-system P is a collection of subsets of Σ that is closed under finitely many intersections, and a Dynkin system (or λ-system) D is a collection of subsets of Σ that contains Σ and is closed under complement and under countable unions of disjoint subsets.

Dynkin’s π-λ theorem says, if P is a π-system and D is a Dynkin system that contains P then the σ-algebra σ(P) generated by P is contained in D. Since certain π-systems are relatively simple classes, it may not be hard to verify that all sets in P enjoy the property under consideration while, on the other hand, showing that the collection D of all subsets with the property is a Dynkin system can also be straightforward. Dynkin’s π-λ Theorem then implies that all sets in σ(P) enjoy the property, avoiding the task of checking it for an arbitrary set in σ(P). One of the most fundamental uses of the π-λ theorem is to show equivalence of separately defined measures or integrals. For example, it is used to equate a probability for a random variable X with the Lebesgue-Stieltjes integral typically associated with computing the probability: ∫ P ∈ (X A) = A F (dx) for all A in the Borel σ-algebra on R,

where F(x) is the cumulative distribution function for X, defined on R, while P is a probability measure, defined on a σ-algebra Σ of subsets of some sample space Ω.

8.2.3 Combining σ-algebras

Suppose {Σα : α ∈ A} is a collection of σ-algebras on a space X.

• The intersection of a collection of σ-algebras is a σ-algebra. To emphasize its character as a σ-algebra, it often is denoted by: 26 CHAPTER 8. SIGMA-ALGEBRA

∧ Σα. α∈A Sketch of Proof: Let Σ∗ denote the intersection. Since X is in every Σα,Σ∗ is not empty. Closure under complement and countable unions for every Σα implies the same must be true for Σ∗. Therefore, Σ∗ is a σ-algebra.

• The union of a collection of σ-algebras is not generally a σ-algebra, or even an algebra, but it generates a σ-algebra known as the join which typically is denoted

( ) ∨ ∪ Σα = σ Σα . α∈A α∈A A π-system that generates the join is { } ∩n P ∈ ∈ A ≥ = Ai : Ai Σαi , αi , n 1 . i=1

Sketch of Proof: By the case n = 1, it is seen that each Σα ⊂ P , so ∪ Σα ⊂ P. α∈A This implies ( ) ∪ σ Σα ⊂ σ(P) α∈A by the definition of a σ-algebra generated by a collection of subsets. On the other hand, ( ) ∪ P ⊂ σ Σα α∈A which, by Dynkin’s π-λ theorem, implies ( ) ∪ σ(P) ⊂ σ Σα . α∈A

8.2.4 σ-algebras for subspaces

Suppose Y is a subset of X and let (X, Σ) be a measurable space.

• The collection {Y ∩ B: B ∈ Σ} is a σ-algebra of subsets of Y. • Suppose (Y, Λ) is a measurable space. The collection {A ⊂ X : A ∩ Y ∈ Λ} is a σ-algebra of subsets of X.

8.2.5 Relation to σ-ring

A σ-algebra Σ is just a σ-ring that contains the universal set X.[3] A σ-ring need not be a σ-algebra, as for example measurable subsets of zero Lebesgue measure in the real line are a σ-ring, but not a σ-algebra since the real line has infinite measure and thus cannot be obtained by their countable union. If, instead of zero measure, one takes measurable subsets of finite Lebesgue measure, those are a ring but not a σ-ring, since the real line can be obtained by their countable union yet its measure is not finite. 8.3. EXAMPLES 27

8.2.6 Typographic note

σ-algebras are sometimes denoted using calligraphic capital letters, or the Fraktur typeface. Thus (X, Σ) may be denoted as (X, F) or (X, F) .

8.3 Examples

8.3.1 Simple set-based examples

Let X be any set.

• The family consisting only of the empty set and the set X, called the minimal or trivial σ-algebra over X. • The power set of X, called the discrete σ-algebra. • The collection {∅, A, Ac, X} is a simple σ-algebra generated by the subset A. • The collection of subsets of X which are countable or whose complements are countable is a σ-algebra (which is distinct from the power set of X if and only if X is uncountable). This is the σ-algebra generated by the singletons of X. Note: “countable” includes finite or empty. • The collection of all unions of sets in a countable partition of X is a σ-algebra.

8.3.2 Stopping time sigma-algebras

A stopping time τ can define a σ -algebra Fτ , the so-called stopping time sigma-algebra, which in a filtered probability space describes the information up to the random time τ in the sense that, if the filtered probability space is interpreted as a random experiment, the maximum information that can be found out about the experiment from arbitrarily often [4] repeating it until the time τ is Fτ .

8.4 σ-algebras generated by families of sets

8.4.1 σ-algebra generated by an arbitrary family

Let F be an arbitrary family of subsets of X. Then there exists a unique smallest σ-algebra which contains every set in F (even though F may or may not itself be a σ-algebra). It is, in fact, the intersection of all σ-algebras containing F. (See intersections of σ-algebras above.) This σ-algebra is denoted σ(F) and is called the σ-algebra generated by F. For a simple example, consider the set X = {1, 2, 3}. Then the σ-algebra generated by the single subset {1} is σ({{1}}) = {∅, {1}, {2, 3}, {1, 2, 3}}. By an abuse of notation, when a collection of subsets contains only one element, A, one may write σ(A) instead of σ({A}); in the prior example σ({1}) instead of σ({{1}}). Indeed, using σ(A1, A2, ...) to mean σ({A1, A2, ...}) is also quite common. There are many families of subsets that generate useful σ-algebras. Some of these are presented here.

8.4.2 σ-algebra generated by a function

If f is a function from a set X to a set Y and B is a σ-algebra of subsets of Y, then the σ-algebra generated by the function f, denoted by σ(f), is the collection of all inverse images f−1(S) of the sets S in B. i.e.

σ(f) = {f −1(S) | S ∈ B}.

A function f from a set X to a set Y is measurable with respect to a σ-algebra Σ of subsets of X if and only if σ(f) is a subset of Σ. 28 CHAPTER 8. SIGMA-ALGEBRA

One common situation, and understood by default if B is not specified explicitly, is when Y is a metric or topological space and B is the collection of Borel sets on Y. If f is a function from X to Rn then σ(f) is generated by the family of subsets which are inverse images of inter- vals/rectangles in Rn:

( ) −1 σ(f) = σ {f ((a1, b1] × · · · × (an, bn]) : ai, bi ∈ R} . A useful property is the following. Assume f is a measurable map from (X,ΣX) to (S,ΣS) and g is a measurable map from (X,ΣX) to (T,ΣT). If there exists a measurable function h from T to S such that f(x) = h(g(x)) then σ(f) ⊂ σ(g). If S is finite or countably infinite or if (S,ΣS) is a standard Borel space (e.g., a separable complete metric space with its associated Borel sets) then the converse is also true.[5] Examples of standard Borel spaces include Rn with its Borel sets and R∞ with the cylinder σ-algebra described below.

8.4.3 Borel and Lebesgue σ-algebras

An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). Note that this σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets. On the Euclidean space Rn, another σ-algebra is of importance: that of all Lebesgue measurable sets. This σ-algebra contains more sets than the Borel σ-algebra on Rn and is preferred in integration theory, as it gives a complete measure space.

8.4.4 Product σ-algebra

Let (X1, Σ1) and (X2, Σ2) be two measurable spaces. The σ-algebra for the corresponding product space X1 × X2 is called the product σ-algebra and is defined by

Σ1 × Σ2 = σ({B1 × B2 : B1 ∈ Σ1,B2 ∈ Σ2}).

Observe that {B1 × B2 : B1 ∈ Σ1,B2 ∈ Σ2} is a π-system. The Borel σ-algebra for Rn is generated by half-infinite rectangles and by finite rectangles. For example,

n B(R ) = σ ({(−∞, b1] × · · · × (−∞, bn]: bi ∈ R}) = σ ({(a1, b1] × · · · × (an, bn]: ai, bi ∈ R}) . For each of these two examples, the generating family is a π-system.

8.4.5 σ-algebra generated by cylinder sets

Suppose

X ⊂ RT = {f : f(t) ∈ R, t ∈ T} is a set of real-valued functions. Let B(R) denote the Borel subsets of R.A cylinder subset of X is a finitely restricted set defined as

{ ∈ ∈ ≤ ≤ } Ct1,...,tn (B1,...,Bn) = f X : f(ti) Bi, 1 i n . Each

{ ∈ B R ≤ ≤ } Ct1,...,tn (B1,...,Bn): Bi ( ), 1 i n 8.5. SEE ALSO 29

is a π-system that generates a σ-algebra Σt1,...,tn . Then the family of subsets

∪∞ ∪ F X = Σt1,...,tn

n=1 ti∈T,i≤n is an algebra that generates the cylinder σ-algebra for X. This σ-algebra is a subalgebra of the Borel σ-algebra determined by the product topology of RT restricted to X. An important special case is when T is the set of natural numbers and X is a set of real-valued sequences. In this case, it suffices to consider the cylinder sets

∞ Cn(B1,...,Bn) = (B1 × · · · × Bn × R ) ∩ X = {(x1, x2, . . . , xn, xn+1,... ) ∈ X : xi ∈ Bi, 1 ≤ i ≤ n}, for which

Σn = σ({Cn(B1,...,Bn): Bi ∈ B(R), 1 ≤ i ≤ n})

is a non-decreasing sequence of σ-algebras.

8.4.6 σ-algebra generated by random variable or vector

Suppose (Ω, Σ, P) is a probability space. If Y :Ω → Rn is measurable with respect to the Borel σ-algebra on Rn then Y is called a random variable (n = 1) or random vector (n ≥ 1). The σ-algebra generated by Y is

σ(Y ) = {Y −1(A): A ∈ B(Rn)}.

8.4.7 σ-algebra generated by a stochastic process

Suppose (Ω, Σ, P) is a probability space and RT is the set of real-valued functions on T . If Y :Ω → X ⊂ RT is measurable with respect to the cylinder σ-algebra σ(FX ) (see above) for X then Y is called a stochastic process or random process. The σ-algebra generated by Y is

{ } −1 −1 σ(Y ) = Y (A): A ∈ σ(FX ) = σ({Y (A): A ∈ FX }),

the σ-algebra generated by the inverse images of cylinder sets.

8.5 See also

• Join (sigma algebra)

• Measurable function

• Sample space

• Separable sigma algebra

• Sigma ring

• Sigma additivity 30 CHAPTER 8. SIGMA-ALGEBRA

8.6 References

[1] Billingsley, Patrick (2012). Probability and Measure (Anniversary ed.). Wiley. ISBN 978-1118122372.

[2] Rudin, Walter (1987). Real & Complex Analysis. McGraw-Hill. ISBN 0-07-054234-1.

[3] Vestrup, Eric M. (2009). The Theory of Measures and Integration. John Wiley & Sons. p. 12. ISBN 9780470317952.

[4] Fischer, Tom (2013). “On simple representations of stopping times and stopping time sigma-algebras”. Statistics and Probability Letters 83 (1): 345–349. doi:10.1016/j.spl.2012.09.024.

[5] Kallenberg, Olav (2001). Foundations of Modern Probability (2nd ed.). Springer. p. 7. ISBN 0-387-95313-2.

8.7 External links

• Hazewinkel, Michiel, ed. (2001), “Algebra of sets”, Encyclopedia of Mathematics, Springer, ISBN 978-1- 55608-010-4

• Sigma Algebra from PlanetMath. Chapter 9

Sigma-ring

In mathematics, a nonempty collection of sets is called a σ-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.

9.1 Formal definition

Let R be a nonempty collection of sets. Then R is a σ-ring if:

∪ ∞ ∈ R ∈ R ∈ N 1. n=1 An if An for all n

2. A \ B ∈ R if A, B ∈ R

9.2 Properties

From these two properties we immediately see that

∩ ∞ ∈ R ∈ R ∈ N n=1 An if An for all n

∩∞ \ ∪∞ \ This is simply because n=1An = A1 n=1(A1 An) .

9.3 Similar concepts

If the first property is weakened to closure under finite union (i.e., A∪B ∈ R whenever A, B ∈ R ) but not countable union, then R is a ring but not a σ-ring.

9.4 Uses

σ-rings can be used instead of σ-fields (σ-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every σ-field is also a σ-ring, but a σ-ring need not be a σ-field. A σ-ring R that is a collection of subsets of X induces a σ-field for X . Define A to be the collection of all subsets of X that are elements of R or whose complements are elements of R . Then A is a σ-field over the set X . In fact A is the minimal σ-field containing R since it must be contained in every σ-field containing R .

31 32 CHAPTER 9. SIGMA-RING

9.5 See also

• Delta ring

• Ring of sets • Sigma field

9.6 References

• Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses σ-rings in development of Lebesgue theory. Chapter 10

Universal algebra

Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples (“models”) of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes “the theory of groups” as an object of study.

10.1 Basic idea

In universal algebra, an algebra (or algebraic structure) is a set A together with a collection of operations on A. An n-ary operation on A is a function that takes n elements of A and returns a single element of A. Thus, a 0-ary operation (or nullary operation) can be represented simply as an element of A, or a constant, often denoted by a letter like a. A 1-ary operation (or unary operation) is simply a function from A to A, often denoted by a symbol placed in front of its argument, like ~x. A 2-ary operation (or binary operation) is often denoted by a symbol placed between its arguments, like x ∗ y. Operations of higher or unspecified arity are usually denoted by function symbols, with the arguments placed in parentheses∧ and separated by commas, like f(x,y,z) or f(x1,...,xn). Some researchers allow infinitary operations, such as α∈J xα where J is an infinite index set, thus leading into the algebraic theory of complete lattices. One way of talking about an algebra, then, is by referring to it as an algebra of a certain type Ω , where Ω is an ordered sequence of natural numbers representing the arity of the operations of the algebra.

10.1.1 Equations

After the operations have been specified, the nature of the algebra can be further limited by axioms, which in universal algebra often take the form of identities, or equational laws. An example is the associative axiom for a binary operation, which is given by the equation x ∗ (y ∗ z) = (x ∗ y) ∗ z. The axiom is intended to hold for all elements x, y, and z of the set A.

10.2 Varieties

Main article: Variety (universal algebra)

An algebraic structure that can be defined by identities is called a variety, and these are sufficiently important that some authors consider varieties the only object of study in universal algebra, while others consider them an object. Restricting one’s study to varieties rules out:

• Predicate logic, notably quantification, including universal quantification ( ∀ ), except before an equation, and existential quantification ( ∃ )

• All relations except equality, in particular inequalities, both a ≠ b and order relations

33 34 CHAPTER 10. UNIVERSAL ALGEBRA

In this narrower definition, universal algebra can be seen as a special branch of model theory, typically dealing with structures having operations only (i.e. the type can have symbols for functions but not for relations other than equality), and in which the language used to talk about these structures uses equations only. Not all algebraic structures in a wider sense fall into this scope. For example ordered groups are not studied in mainstream universal algebra because they involve an ordering relation. A more fundamental restriction is that universal algebra cannot study the class of fields, because there is no type (a.k.a. signature) in which all field laws can be written as equations (inverses of elements are defined for all non-zero elements in a field, so inversion cannot simply be added to the type). One advantage of this restriction is that the structures studied in universal algebra can be defined in any category that has finite products. For example, a topological group is just a group in the category of topological spaces.

10.2.1 Examples

Most of the usual algebraic systems of mathematics are examples of varieties, but not always in an obvious way – the usual definitions often involve quantification or inequalities.

Groups

To see how this works, let’s consider the definition of a group. Normally a group is defined in terms of a single binary operation ∗, subject to these axioms:

• Associativity (as in the previous section): x ∗ (y ∗ z) = (x ∗ y) ∗ z; formally: ∀x,y,z. x∗(y∗z)=(x∗y)∗z. • Identity element: There exists an element e such that for each element x, e ∗ x = x = x ∗ e; formally: ∃e ∀x. e∗x=x=x∗e. • Inverse element: It can easily be seen that the identity element is unique. If this unique identity element is denoted by e then for each x, there exists an element i such that x ∗ i = e = i ∗ x; formally: ∀x ∃i. x∗i=e=i∗x.

(Some authors also use an axiom called "closure", stating that x ∗ y belongs to the set A whenever x and y do. But from a universal algebraist’s point of view, that is already implied by calling ∗ a binary operation.) This definition of a group is problematic from the point of view of universal algebra. The reason is that the axioms of the identity element and inversion are not stated purely in terms of equational laws but also have clauses involving the phrase “there exists ... such that ...”. This is inconvenient; the list of group properties can be simplified to universally quantified equations by adding a nullary operation e and a unary operation ~ in addition to the binary operation ∗. Then list the axioms for these three operations as follows:

• Associativity: x ∗ (y ∗ z) = (x ∗ y) ∗ z. • Identity element: e ∗ x = x = x ∗ e; formally: ∀x. e∗x=x=x∗e. • Inverse element: x ∗ (~x) = e = (~x) ∗ x formally: ∀x. x∗~x=e=~x∗x.

(Of course, we usually write "x−1" instead of "~x", which shows that the notation for operations of low arity is not always as given in the second paragraph.) What has changed is that in the usual definition there are:

• a single binary operation (signature (2)) • 1 equational law (associativity) • 2 quantified laws (identity and inverse)

...while in the universal algebra definition there are

• 3 operations: one binary, one unary, and one nullary (signature (2,1,0)) 10.3. BASIC CONSTRUCTIONS 35

• 3 equational laws (associativity, identity, and inverse) • no quantified laws (except for outermost universal quantifiers which are allowed in varieties)

It is important to check that this really does capture the definition of a group. The reason that it might not is that specifying one of these universal groups might give more information than specifying one of the usual kind of group. After all, nothing in the usual definition said that the identity element e was unique; if there is another identity element e', then it is ambiguous which one should be the value of the nullary operator e. Proving that it is unique is a common beginning exercise in classical group theory textbooks. The same thing is true of inverse elements. So, the universal algebraist’s definition of a group is equivalent to the usual definition. At first glance this is simply a technical difference, replacing quantified laws with equational laws. However, it has immediate practical consequences – when defining a group object in category theory, where the object in question may not be a set, one must use equational laws (which make sense in general categories), and cannot use quantified laws (which do not make sense, as objects in general categories do not have elements). Further, the perspective of universal algebra insists not only that the inverse and identity exist, but that they be maps in the category. The basic example is of a topological group – not only must the inverse exist element-wise, but the inverse map must be continuous (some authors also require the identity map to be a closed inclusion, hence cofibration, again referring to properties of the map).

10.3 Basic constructions

We assume that the type, Ω , has been fixed. Then there are three basic constructions in universal algebra: homo- morphic image, subalgebra, and product. A homomorphism between two algebras A and B is a function h: A → B from the set A to the set B such that, for every operation fA of A and corresponding fB of B (of arity, say, n), h(fA(x1,...,xn)) = fB(h(x1),...,h(xn)). (Sometimes the subscripts on f are taken off when it is clear from context which algebra your function is from.) For example, if e is a constant (nullary operation), then h(eA) = eB. If ~ is a unary operation, then h(~x) = ~h(x). If ∗ is a binary operation, then h(x ∗ y) = h(x) ∗ h(y). And so on. A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under the entry Homomorphism. In particular, we can take the homomorphic image of an algebra, h(A). A subalgebra of A is a subset of A that is closed under all the operations of A. A product of some set of algebraic structures is the cartesian product of the sets with the operations defined coordinatewise.

10.4 Some basic theorems

• The isomorphism theorems, which encompass the isomorphism theorems of groups, rings, modules, etc. • Birkhoff’s HSP Theorem, which states that a class of algebras is a variety if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products.

10.5 Motivations and applications

In addition to its unifying approach, universal algebra also gives deep theorems and important examples and coun- terexamples. It provides a useful framework for those who intend to start the study of new classes of algebras. It can enable the use of methods invented for some particular classes of algebras to other classes of algebras, by recasting the methods in terms of universal algebra (if possible), and then interpreting these as applied to other classes. It has also provided conceptual clarification; as J.D.H. Smith puts it, “What looks messy and complicated in a particular framework may turn out to be simple and obvious in the proper general one.” In particular, universal algebra can be applied to the study of monoids, rings, and lattices. Before universal algebra came along, many theorems (most notably the isomorphism theorems) were proved separately in all of these fields, but with universal algebra, they can be proven once and for all for every kind of algebraic system. The 1956 paper by Higgins referenced below has been well followed up for its framework for a range of particular algebraic systems, while his 1963 paper is notable for its discussion of algebras with operations which are only partially 36 CHAPTER 10. UNIVERSAL ALGEBRA defined, typical examples for this being categories and groupoids. This leads on to the subject of higher-dimensional algebra which can be defined as the study of algebraic theories with partial operations whose domains are defined under geometric conditions. Notable examples of these are various forms of higher-dimensional categories and groupoids.

10.6 Generalizations

Further information: Category theory, Operad theory and Partial algebra

A more generalised programme along these lines is carried out by category theory. Given a list of operations and axioms in universal algebra, the corresponding algebras and homomorphisms are the objects and morphisms of a category. Category theory applies to many situations where universal algebra does not, extending the reach of the theorems. Conversely, many theorems that hold in universal algebra do not generalise all the way to category theory. Thus both fields of study are useful. A more recent development in category theory that generalizes operations is operad theory – an operad is a set of operations, similar to a universal algebra. Another development is partial algebra where the operators can be partial functions. An important generalization of universal algebra theory is model theory, which is sometimes described as “universal theory + logic”.

10.7 History

In Alfred North Whitehead's book A Treatise on Universal Algebra, published in 1898, the term universal algebra had essentially the same meaning that it has today. Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself.[1] At the time structures such as Lie algebras and hyperbolic quaternions drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review Alexander Macfarlane wrote: “The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures.” At the time George Boole's algebra of logic made a strong counterpoint to ordinary number algebra, so the term “universal” served to calm strained sensibilities. Whitehead’s early work sought to unify quaternions (due to Hamilton), Grassmann's Ausdehnungslehre, and Boole’s algebra of logic. Whitehead wrote in his book:

“Such algebras have an intrinsic value for separate detailed study; also they are worthy of comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular. The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge.”[2]

Whitehead, however, had no results of a general nature. Work on the subject was minimal until the early 1930s, when Garrett Birkhoff and Øystein Ore began publishing on universal algebras. Developments in metamathematics and category theory in the 1940s and 1950s furthered the field, particularly the work of Abraham Robinson, Alfred Tarski, Andrzej Mostowski, and their students (Brainerd 1967). In the period between 1935 and 1950, most papers were written along the lines suggested by Birkhoff’s papers, dealing with free algebras, congruence and subalgebra lattices, and homomorphism theorems. Although the development of mathematical logic had made applications to algebra possible, they came about slowly; results published by Anatoly Maltsev in the 1940s went unnoticed because of the war. Tarski’s lecture at the 1950 International Congress of Mathematicians in Cambridge ushered in a new period in which model-theoretic aspects were developed, mainly by Tarski himself, as well as C.C. Chang, Leon Henkin, Bjarni Jónsson, Roger Lyndon, and others. In the late 1950s, Edward Marczewski[3] emphasized the importance of free algebras, leading to the publication of more than 50 papers on the algebraic theory of free algebras by Marczewski himself, together with Jan Mycielski, Władysław Narkiewicz, Witold Nitka, J. Płonka, S. Świerczkowski, K. Urbanik, and others. 10.8. SEE ALSO 37

10.8 See also

• Graph algebra • Homomorphism • Lattice theory • Signature • Term algebra • Variety • Clone • Universal algebraic geometry • Model theory

10.9 Footnotes

[1] Grätzer, George. Universal Algebra, Van Nostrand Co., Inc., 1968, p. v. [2] Quoted in Grätzer, George. Universal Algebra, Van Nostrand Co., Inc., 1968. [3] Marczewski, E. “A general scheme of the notions of independence in mathematics.” Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 6 (1958), 731–736.

10.10 References

• Bergman, George M., 1998. An Invitation to General Algebra and Universal Constructions (pub. Henry Helson, 15 the Crescent, Berkeley CA, 94708) 398 pp. ISBN 0-9655211-4-1. • Birkhoff, Garrett, 1946. Universal algebra. Comptes Rendus du Premier Congrès Canadien de Mathématiques, University of Toronto Press, Toronto, pp. 310–326. • Brainerd, Barron, Aug–Sep 1967. Review of Universal Algebra by P. M. Cohn. American Mathematical Monthly, 74(7): 878–880. • Burris, Stanley N., and H.P. Sankappanavar, 1981. A Course in Universal Algebra Springer-Verlag. ISBN 3-540-90578-2 Free online edition. • Cohn, Paul Moritz, 1981. Universal Algebra. Dordrecht, Netherlands: D.Reidel Publishing. ISBN 90-277- 1213-1 (First published in 1965 by Harper & Row) • Freese, Ralph, and Ralph McKenzie, 1987. Commutator Theory for Congruence Modular Varieties, 1st ed. London Mathematical Society Lecture Note Series, 125. Cambridge Univ. Press. ISBN 0-521-34832-3. Free online second edition. • Grätzer, George, 1968. Universal Algebra D. Van Nostrand Company, Inc. • Higgins, P. J. Groups with multiple operators. Proc. London Math. Soc. (3) 6 (1956), 366–416. • Higgins, P.J., Algebras with a scheme of operators. Mathematische Nachrichten (27) (1963) 115–132. • Hobby, David, and Ralph McKenzie, 1988. The Structure of Finite Algebras American Mathematical Society. ISBN 0-8218-3400-2. Free online edition. • Jipsen, Peter, and Henry Rose, 1992. Varieties of Lattices, Lecture Notes in Mathematics 1533. Springer Verlag. ISBN 0-387-56314-8. Free online edition. • Pigozzi, Don. General Theory of Algebras. • Smith, J.D.H., 1976. Mal'cev Varieties, Springer-Verlag. • Whitehead, Alfred North, 1898. A Treatise on Universal Algebra, Cambridge. (Mainly of historical interest.) 38 CHAPTER 10. UNIVERSAL ALGEBRA

10.11 External links

• Algebra Universalis—a journal dedicated to Universal Algebra. 10.12. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 39

10.12 Text and image sources, contributors, and licenses

10.12.1 Text

• Algebra of sets Source: https://en.wikipedia.org/wiki/Algebra_of_sets?oldid=660214875 Contributors: The Anome, AugPi, Charles Matthews, WhisperToMe, Wile E. Heresiarch, Tobias Bergemann, Macrakis, Doshell, Discospinster, Esse~enwiki, Slipstream, Paul Au- gust, Oleg Alexandrov, Woohookitty, Isnow, Salix alba, Juan Marquez, Mathbot, Splintercellguy, Trovatore, Arthur Rubin, Gilliam, Bluebot, Javalenok, Byelf2007, Jackzhp, Kupirijo, Josephpetty100, Policron, The enemies of god, Alex10023, Jamelan, Tcamps42, An- chor Link Bot, Hans Adler, Addbot, Legobot, TaBOT-zerem, Matt Popat, Bluerasberry, Corruptcopper, Saeidpourbabak, Yahia.barie, Specs112, MegaSloth, DASHBot, Set theorist, ClueBot NG, MerlIwBot, DonMTobin, Helpful Pixie Bot, Daviddwd, Palltrast, Chris- Gualtieri, Freeze S, Jochen Burghardt, YiFeiBot, Eniacpx, Plcarmelbiron and Anonymous: 41 • Join (sigma algebra) Source: https://en.wikipedia.org/wiki/Join_(sigma_algebra)?oldid=587405272 Contributors: Tsirel, Linas and Anonymous: 1 • Measurable function Source: https://en.wikipedia.org/wiki/Measurable_function?oldid=684281044 Contributors: AxelBoldt, Zundark, Miguel~enwiki, Imran, Silverfish, Charles Matthews, Dysprosia, Jitse Niesen, Fibonacci, Robbot, Weialawaga~enwiki, Tosha, Giftlite, Rs2, Fropuff, CSTAR, Maneesh, Vivacissamamente, Paul August, Msh210, Oleg Alexandrov, Linas, OdedSchramm, Mandarax, MarSch, Salix alba, Mike Segal, Chobot, RussBot, GeeJo, Dan131m, Zvika, SmackBot, Gala.martin, Chungc, Richard L. Peterson, Vanished user v8n3489h3tkjnsdkq30u3f, Pascal.Tesson, Thijs!bot, GiM, Salgueiro~enwiki, Takwan, STBot, Jahredtobin, Jka02, AlleborgoBot, SieBot, COBot, Rinconsoleao, Dingenis, CàlculIntegral, Addbot, Loewepeter, PV=nRT, Yobot, Ht686rg90, Calle, Bdmy, GrouchoBot, Undsoweiter, Sławomir Biały, Mjs1991, Walterfm, Ron asquith, EmausBot, ZéroBot, Parodi, Helpful Pixie Bot, Brirush, LHSPhantom, Shorbowman and Anonymous: 36 • Pi system Source: https://en.wikipedia.org/wiki/Pi_system?oldid=681502917 Contributors: Charles Matthews, Salix alba, Wavelength, Spacepotato, Dmharvey, Stifle, Mattroberts, Ru elio, Vanish2, Sullivan.t.j, David Eppstein, TXiKiBoT, Kinrayfchen, Addbot, Aroch, Yobot, Hairer, DrilBot, Helpful Pixie Bot, BG19bot, Odaniel1, Savick01, Bwangaa, Egreif1, DarenCline, Cyrus Cousins and Anonymous: 10 • Sample space Source: https://en.wikipedia.org/wiki/Sample_space?oldid=669667565 Contributors: Lee Daniel Crocker, Zundark, Patrick, Tomi, Fwappler, Andres, Bjcairns, Dysprosia, Mountain, Giftlite, Jason Quinn, Discospinster, MisterSheik, El C, Obradovic Goran, Jum- buck, Msh210, Oleg Alexandrov, INic, Linas, Isnow, Graham87, FlaBot, Siddhant, B-Con, SmackBot, Snielsen, Gelingvistoj, Nbarth, KaiserbBot, The Phantom Editor, Thijs!bot, Minhtung91, Nyttend, J.delanoy, VolkovBot, Daviddoria, Paradoctor, Philadams, Mel- combe, ClueBot, Justin W Smith, NuclearWarfare, Wildscop, Kaiba, Aitias, Addbot, 2rock, Gail, Yobot, TaBOT-zerem, Niout, Ciphers, Ayda D, Capricorn42, Erik9bot, Thehelpfulbot, Yuanfangdelang, RandomDSdevel, HRoestBot, Gnathan87, Seahorseruler, RjwilmsiBot, EmausBot, Tommy2010, ClueBot NG, MerlIwBot, BG19bot, HilberTraum, ChrisGualtieri, TheJJJunk, Mr. Guye, Pchisq, Bbpfirst, Bryanrutherford0 and Anonymous: 57 • Separable sigma algebra Source: https://en.wikipedia.org/wiki/Separable_sigma_algebra?oldid=654658454 Contributors: Tobias Berge- mann, Tsirel, Oleg Alexandrov, Salix alba, CBM, Thijs!bot, Addbot, Topology Expert, Yobot, AnomieBOT, Erik9bot, RedBot, Zfeinst, Qetuth, Khazar2 and Anonymous: 3 • Sigma additivity Source: https://en.wikipedia.org/wiki/Sigma_additivity?oldid=670579000 Contributors: Michael Hardy, Chinju, An- dres, Charles Matthews, Fibonacci, MathMartin, Vivacissamamente, Paul August, Rgdboer, Blotwell, Oleg Alexandrov, Linas, Salix alba, -A. Pichler, WISo, TM-77, R'n'B, Mikemtha, Daniele.tampieri, Dingenis, Addbot, SamatBot, Luckas-bot, Yobot, RandomDS ,דניאל צבי devel, 777sms, John of Reading, ZéroBot, Quondum, WoldaCZ and Anonymous: 15 • Sigma-algebra Source: https://en.wikipedia.org/wiki/Sigma-algebra?oldid=694395546 Contributors: AxelBoldt, Zundark, Tarquin, Iwn- bap, Miguel~enwiki, Michael Hardy, Chinju, Karada, Stevan White, Charles Matthews, Dysprosia, Vrable, AndrewKepert, Fibonacci, Robbot, Romanm, Ruakh, Giftlite, Lethe, MathKnight, Mboverload, Gubbubu, Gauss, Barnaby dawson, Vivacissamamente, William Elliot, ArnoldReinhold, Paul August, Bender235, Zaslav, Elwikipedista~enwiki, MisterSheik, EmilJ, SgtThroat, Jung dalglish, Tsirel, Passw0rd, Msh210, Jheald, Cmapm, Ultramarine, Oleg Alexandrov, Linas, Graham87, Salix alba, FlaBot, Mathbot, Jrtayloriv, Chobot, Jayme, YurikBot, Lucinos~enwiki, Archelon, Trovatore, Mindthief, Solstag, Crasshopper, Dinno~enwiki, Nielses, SmackBot, Melchoir, JanusDC, Object01, Dan Hoey, MalafayaBot, RayAYang, Nbarth, DHN-bot~enwiki, Javalenok, Gala.martin, Henning Makholm, Lam- biam, Dbtfz, Jim.belk, Mets501, Stotr~enwiki, Madmath789, CRGreathouse, CBM, David Cooke, Mct mht, Blaisorblade, Xanthar- ius, , Thijs!bot, Escarbot, Keith111, Forgetfulfunctor, Quentar~enwiki, MSBOT, Magioladitis, RogierBrussee, Paartha, Joeabauer, Hippasus, Policron, Cerberus0, Digby Tantrum, Jmath666, Alfredo J. Herrera Lago, StevenJohnston, Ocsenave, Tcamps42, SieBot, Mel- combe, MicoFilós~enwiki, Andrewbt, The Thing That Should Not Be, Mild Bill Hiccup, BearMachine, 1ForTheMoney, DumZiBoT, Ad- dbot, Luckas-bot, Yobot, Li3939108, Amirmath, Godvjrt, Xqbot, RibotBOT, Charvest, FrescoBot, BrideOfKripkenstein, AstaBOTh15, Stpasha, RedBot, Soumyakundu, Wikiborg4711, Stj6, TjBot, Max139, KHamsun, Rafi5749, ClueBot NG, Thegamer 298, QuarkyPi, Brad7777, AntanO, Shalapaws, Crawfoal, Dexbot, Y256, Jochen Burghardt, A koyee314, Limit-theorem, Mark viking, NumSPDE, Moyaccercchi, Killaman slaughtermaster, DarenCline, 5D2262B74 and Anonymous: 84 • Sigma-ring Source: https://en.wikipedia.org/wiki/Sigma-ring?oldid=667498132 Contributors: Charles Matthews, Touriste, Salix alba, Maxbaroi, Konradek, Keith111, Addbot, Luckas-bot, Xqbot, ZéroBot, Zephyrus Tavvier, Jim Sukwutput, Brirush, DarenCline, Gana- tuiyop and Anonymous: 3 • Universal algebra Source: https://en.wikipedia.org/wiki/Universal_algebra?oldid=692759452 Contributors: AxelBoldt, Bryan Derksen, Zundark, Andre Engels, Toby~enwiki, Toby Bartels, Youandme, Michael Hardy, GTBacchus, Andres, Revolver, Charles Matthews, Dysprosia, Aleph4, Robbot, Fredrik, Sanders muc, Kowey, Fuelbottle, Giftlite, APH, Sam Hocevar, AlexChurchill, Zaslav, Tompw, Rgdboer, EmilJ, Chasmo, AshtonBenson, Msh210, ABCD, Linas, Mindmatrix, Smmurphy, Isnow, Magidin, Jrtayloriv, Wavelength, Hairy Dude, Chaos, Wiki alf, Arthur Rubin, RonnieBrown, SmackBot, Gilliam, El Fahno, Alink, Nbarth, Zvar, Spakoj~enwiki, Henning Makholm, WillowW, HStel, Sam Staton, Sadeghd, Rlupsa, Knotwork, JAnDbot, Sean Tilson, Twisted86, David Eppstein, JaGa, Pavel Jelínek, Trusilver, TheSeven, JohnBlackburne, AllS33ingI, Popopp, Synthebot, Nicks221, SieBot, Sneakfast, Tkeu, Excirial, He7d3r, Hans Adler, Kaba3, Algebran, Addbot, Delaszk, Loupeter, Legobot, Yobot, Ptbotgourou, AnomieBOT, Materialscientist, RibotBOT, Oursipan, Thehelpfulbot, Ilovegrouptheory, D stankov, Yaddie, Rausch, Eivuokko, Nascar1996, GoingBatty, Quondum, ClueBot NG, Bezik, Frietjes, MerlIwBot, Beaumont877, Brad7777, Duxwing, Jochen Burghardt, NQ, JMP EAX, JohnAGough and Anonymous: 64 40 CHAPTER 10. 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