ECON5155 Autumn 2012: Notes and Curriculum

Universitetet i Oslo / Økonomisk institutt / NCF; this version December 3, 2012 ECON5155 autumn 2012: Notes and curriculum

This document summarizes key concepts and learning outcome goals for the course, including typical problems you will be expected to be able to solve. It will be more elaborate on topics where there is no set literature (then this document serves as a reference).

Now the exam is open book, so you are allowed to bring any printed or written resource. That of course includes sources which have not been subject to ordinary scientific quality assurance – like your own notes, like this note or like any Wikipedia article. However, just like other mathematics exams, you will be expected to show your calculations from known results. To avoid the issue of trustworthiness of sources which can at any time be changed by anyone without any scientific review (e.g. Wikipedia), I have compiled and attached a few such articles which I have checked, and officially OK’ed for the purposes of this course.

0 The reading list

The reading list is taken from the following:

• This document (all of it) except the attachment from free sources, which is included as a service.

• Tom Lindstrøm’s lecture notes to MAT2400, available at http://www.uio.no/studier/ emner/matnat/math/MAT2400/v12/MathAnalBook-Tom.pdf (with errata here). All references to this version (note that the 2011 version had completely different enumera- tion).

• Atle Seierstad: «Stochastic Control in Discrete and Continuous Time», Springer. Avail- able online for UiO login, as of now at http://link.springer.com/book/10.1007/ 978-0-387-76617-1/.

• All the problems assigned are curriculum.

From Lindstrøm:

• Preliminaries (Chapter 1)

• Metric spaces (Chapter 2)

• Regarding Chapter 3: – The required topics concerning convergence is covered in this note – in the setup of general topology as well – and Lindstrøm section 3.2 is on the reading list to the extent it supports those. – The required topics concerning convergence is covered in this note – again, in the setup of general topology as well – and Lindstrøm section 3.1 and its special treatment of uniform continuity is only supplementary reading.

1 – Lindstrøm section 3.3: the topic of vector spaces, and thus function spaces, is itself on curriculum, and you must know the space of continuous functions, and the supremum metric (indeed, the supremum norm as well). – Lindstrøm section 3.4 is an application of the Banach ﬁxed-point theorem, and was assigned as a problem. – The rest of Lindstrøm chapter 3 has not been covered per se.

• Regarding Chapter 4: – The concepts of limsup / liminf are covered (they are preliminary concepts). – Vector spaces and normed spaces (section 4.5): To read: to and including the Remark on top page 101. You are not required to «recognize» vector spaces by applying each and every part of the definition, but you should be able to see that e.g. the space of positive functions is not a vector space (point out «−f»). You are required to recognize norms using the definition, to test a given functional to check whether it is a norm or not – just like you are required to recognize a metric in the same manner; you should in particular recognize the L1, L2 and L∞ norms just as you should recognize the taxicab, Euclidean and uniform metric. We have however not covered the (Minkowski, Seierstad formula (5.30)) triangle inequality for other Lp spaces, and you can/shall take as granted without justification that the p-norms are indeed norms for any p ≥ 1. – Inner products have only been mentioned as application. It might be worth to know that the inner product with itself is the square of the 2-norm.

• Regarding Chapter 5: 1 Note that Lindstrøm restricts himself to the Lebesgue measure (i.e. length in R , area in 2 R etc.) – we have covered more general measures, see e.g. Seierstad chapter 5 below. Also, see this note. – Skip: Outer measures and the construction of measures by way of such; You should however know the fact that measures can be deﬁned by starting with «prototypical sets» (e.g. open intervals for the Borel case). From Seierstad: • From Chapter 1: Sections 1.1–1.2; then 1.5–1.6 as special cases of 1.1–1.2. Furthermore, 1.9 to the extent it covers the problem given on ﬁxed point iteration. Also, refer to section 1.11 for problems.

• Chapter 4, except – page 189 is for the curious – regarding Theorem 4.1, you only need to know that it basically works analogous to the deterministic case (which was assigned as application of the Banach ﬁxed- point theorem) – the proof of the Itô formula – although the basic idea and the limiting transition proving (dB)2 = dt, is an application of convergence in norm.

2 – regarding the conditions for the Dynkin formula, it sufﬁces to know that you must in the general case bound the stopping times and then apply a limit transition; Limit transitions are per se curriculum, but for exam purposes, you are required to perform this exercise only if asked to. – the Black–Scholes formula, and its proof, are not on curriculum. You will however need some of the arguments: the Itô formula and the heat equation. – Girsanov’s theorem page 207. – For optimal control, like for Dynkin’s formula, you need to be aware that there are limit transition issues as time grows, but you are required to carry them out only if asked to. – Skip the last part of section 4.4, from and including subsection «Soft Terminal Restrictions». – For optimal stopping, you are only required to treat the problems as a special case of optimal control (i.e. the special form of HJB, and the reservation for limit transitions still apply) plus you need to be able to apply the C1 ﬁt (taking the validity for granted). – Skip sections 4.6 and 4.7. • Chapter 5, although Minkowski (5.30) has not been covered. • Also, you could use the backmatter with problem solutions.

1 Stochastic dynamic optimization

Reading list here is Seierstad as detailed above.

Sample problems: • As assigned, although I think 1.26 would be a bit lengthy. • Example problem set Problem 1.

2 Metric spaces

Reading list here is Lindstrøm as assigned. You should in addition know that the concepts of open sets and of compactness as used by Lindstrøm, have different-looking deﬁnitions for the general topology case.

Sample problems: As assigned earlier.

3 Topology: the concepts and the ﬁxed-point application

The treatment of topological topics is somewhat involved. Some topological concepts were deﬁned both in the general case and in the metric (and even more specialized, norm-induced) case.

3 3.1 Before we start on topology itself: sequence, subsequence, countability, complement, union and intersection In this part, Lindstrøm Chapter 1 may also be helpful.

1 A sequence {xn}n∈N – also frequently denoted simply by «{xn}» or by «xn» – is a function whose domain is the natural numbers. That is, it is infinite; sometimes we extend «finite» sequences {x1, . . . , xN } (i.e. an N-vector) to infinite by putting xn = xN for n > N.

A sequence {yk} is a subsequence of {xn} if there is some strictly increasing (inﬁnite!) se-

quence {nk} of natural numbers, so that yk = xnk for all k ∈ N.

We speak of a sequence in a given set A if the range of this function is contained in A.

A set is countable if it can be covered by a sequence – notice that some authors require that the set is inﬁnite, in which case the «ﬁnite-or-countable» property is referred to as at most countable.

The complement of a set A, is the set of points not in A. The complement is relative to the 2 space X in question; if X = R, then the complement of [0, ∞) is (−∞, 0], but if X = R it is the plane with the half-line cut out.

To confuse the {xn} notation even more: For an indexed family of points or sets – say, {xi}i∈I , the index set I will not taken to be countable, unless otherwise stated – for example by writing x1, x2,... (this language signiﬁes that if you continue enumerating this way, you will cover them all), or by calling it a «sequence».

The union of a family {Ai} of sets is the set of points belonging to one or more Ai. The intersection is the set of points which belong to all. Notice again that the family need not be over countably many – therefore, do not write e.g. A1 ∪ A2 ∪ ... unless this is justiﬁed! The usual convention when the index set is empty, is to let the union of no sets be the empty set, and the intersection over no sets be the space itself.

Need to know The following properties are preliminaries which you need to know, and which you in an exam situation can use without proof:

• An interval (a, b) ⊆ R can be written as the union [ (a, b) = {(ar, br); ar > a and br < b are rational}.

Furthermore, since there are only countably many rationals (i.e. no more than can be covered by a sequence), the union is actually taken over countably many invervals.

1Observe the potentially confusing notation: a set containing a single point x¯, is denoted {x¯}, so there might sometimes be an ambiguity of whether «{xn}» refers to the singleton set containing the one and only xn, or the set containing them all. Some authors denote sequences with ordinary parentheses, like vectors of inﬁnite dimension (which they are!).

4 Finally, notice that this union (even reduced to countably many as in the previous sentence), can be written: [ (a, b) = {[ar, br]; ar > a and br < b are rational}.

• By passing to complements, the interval [a, b] can be written as the intersection \ [a, b] = {(ar, br); ar < a and br > b are rational} \ = {[ar, br]; ar < a and br > b are rational}.

This is actually a countable intersection, as the rationals are countable.

• Notice that these facts hold without any convergence of ar to a nor br to b being assumed – nor even deﬁned. They are not topological properties.

3.2 Open sets, neighbourhoods and topological sub-bases For a metric space with metric d, an open r-ball around x¯ is the set of x such that d(¯x, x) < r. This is also called an r-neighbourhood of x¯.

In metric spaces, a neigbhourhood of x¯ is usually thought of this way – a small open ball centered at x¯ – and it is common to say that a set U is open if it contains an open ball around each of its points. (And if it has no points: ∅ is open, but it is not a neighbourhood.) Let us modify the wording slightly: • U is open if it is or contains a neighbourhood around each of its points.

Think of the open r-balls (for varying r) as «prototypical» open sets; all other open sets can be generated from those by taking arbitrary unions and ﬁnite intersections. Here we include r = 0 for the empty set, and r = +∞ for the space itself, both sets always considered open. Such a set of «prototypical» open sets we use to generate the opens, is called a sub-base for the topology.

For the general topological concept, we decide on which sets to call «open», subject to the axioms that the empty set ∅ and the space itself are open, that an (arbitrary) union of opens is open, and that a ﬁnite intersection of opens is open. A family of such sets is called a topology on the space, and a space equipped with a topology, is called a topological space. Again, we can start with a family of «prototypical» open sets and generate the other opens from this (after including the empty set and the space itself), and again, we call this family of prototypes a sub-base.

What then about neighbourhoods? A neighbourhood of x¯ is simply an open set containing x (regardless of shape and size). Then again, as in the metric case: a set U is open if and only if it contains or equals a neighbourhood about each of its members.

Examples • The trivial topology (also known as the «indiscrete» topology) has only the empty set and the space as open.

5 • The discrete topology has all sets open. A sub-base generating this topology, is the set of all singletons.

• The usual topology (also referred to as standard topology) on R can be generated by taking the intervals (a, b) as sub-base – then it also sufﬁces to restrict onself to the ones with a, b rational.

• The integers Z with the standard metric (i.e. the one which is standard for R, namely |x − y|) – on Z this is indeed the discrete topology, as the open 1-balls are singletons.

• The set N ∪ {+∞} with the topology deﬁned by the following sub-basis: {n} are open sets when n ∈ N (but {∞} is not!). In addition, the sets {∞} ∪ {n, n + 1, n + 2,... } for any n ∈ N. This way, a neighbourhood of inﬁnity contains «anything large enough».

3.3 Closed sets A set C is closed if its complement is open. Notice that a closed set can be open; the space itself and the empty set are always both open and closed. And there might be others: in the discrete toplogy, all sets are open and therefore all sets are closed.

You are expected to be able to manipulate open and closed sets using for example DeMor- gan’s laws (for sets as presented in Lindstrøm Prop. 1.2.2 – they are sometimes also given in terms of logic). It follows from the axioms of topology that a ﬁnite union of closeds is closed, and an arbitrary intersection of closeds is closed.

Sample problem Consider R (with standard metric), and the sequence

An = (−1/n, 1 + 1/n) ∪ [n, ∞).

Prove that B = A1 ∩ A2 ∩ ... is closed.

3.4 Limits of sequences This course shall only consider limits of sequences (not of for example nets).

We say that a sequence {xn} = {xn}n=1,2,... converges to x¯ if for any neighbourhood U of x¯, there is an N = NU such that xn ∈ U whenever n > N. Since a neighbourhood of x¯ always contains x, the constant sequence (¯x, x,¯ ...) always converges to x¯.

In the metric case, we require each > 0 there is an N = N such that d(¯x, xn) < whenever n > N. This is merely a specialization of the general concept, where we require «-neighbourhood» in place of neighbourhood. It is seemingly a bit restrictive to consider only the open balls where the general concept requires all open sets containing x¯, but in a metric space, any open set around x¯ contains an open ball, so the convergent sequences will be the same. It might be fruitful to think of as an «error margin». The sequence converges to x¯ if it sooner or later enters for good inside the error margin – i.e., no matter what open U 3 x¯, we can always crop off the ﬁrst N terms from the sequence, and the rest will be a subset of U.

6 So, what does this «error margin» interpretation mean in the general topology setting? Pre- cisely the same, except we do not have a distance measure: we have to specify «error margin» as «membership of U». And the choice of topology speciﬁes what U we can require. If 2 a topology τ2 has all the open sets of another topology τ1 has, and then some in addition , then the τ2 case poses a harder test for convergence: there are more open tests to potentially destroy convergence with.

Examples Consider the examples from the open sets subsection 3.2:

• In the trivial topology, any sequence will converge to every point. (We say that this topology does not separate points.)

• The discrete topology, on the other hand, is the hardest test for convergence: a sequence {xn} converges to x¯ if and only if xn =x ¯ for all but ﬁnitely many n.

• The usual topology R corresponds to the usual concept of convergence.

• The integers Z with the usual metric is again discrete, so convergence means convergence in ﬁnitely many steps.

• The set N ∪ {+∞} with the topology generated by the sets {n} ⊂ N and for n ∈ N: {∞} ∪ {n, n + 1, n + 2,... } for any n ∈ N. Convergence to n¯ ∈ N again means convergence in finitely many steps. But for n¯ = +∞, then mn → ∞ if for any prespecified threshold Q, we have mn > Q for all n large enough. Notice how this agrees with the usual «n tends to infinity» concept – let mn = n. This sequence does «tend» to infinity – indeed it converges to +∞. Not diverges, but converges, as ∞ is a point in the space.

Sample problem: Problem 2 in the 2011 sample problem set.

3.5 Compactness / sequential compactness – equivalence in metric spaces A set K is compact if any open covering can be reduced to finite. In more specific terms, K is compact if the following holds: whenever there is a family {Ui}i∈I of open sets so that S i∈I Ui ⊇ K, then there exists a finite selection i1, . . . , iN such that K ⊆ Ui1 ∪ ... ∪ UiN .

A set K is sequentially compact if any sequence {xn} in K has a subsequence which converges in K.

Compactness and sequential compactness are generally different concepts. It is not so in general, that one implies the other. But in metric spaces, they are equivalent (Lindstrøm Theorem 2.6.6). Notice that Lindstrøm uses the sequential compactness deﬁnition and calls it compact – which is correct in his metric spaces setup, but not beyond that.

2 τ2 is then called strictly ﬁner than τ1, which is strictly coarser than τ2 – you do not need to know these terms though.

7 3.6 Continuity In order to deﬁne continuity of a function, we need to specify a topology on its domain X and its range Y. For the -δ deﬁnition for metric spaces, refer to Lindstrøm. It will turn out to be a special case of the following:

For the general topology case, the deﬁnition is as follows: f is continuous at x¯ ∈ X if for all neighbourhoods V of f(¯x), there is a neighbourhood U of x¯ such that f(U) ⊆ V .

This deﬁnition is property (ii) in Lindstrøm Proposition 2.3.8. It uses the image f(U) notation: f(U) is the set of y which equal f(x) for some x in U. Likewise, the inverse image f −1(V ) is the set of those x for which f(x) ∈ V – this deﬁnition does not require any invert- ibility!

We have the same interpretation as for the «error margins»; for f to qualify as «continuous at x¯», it should be so that you can force it within any error margin V about f(¯x) by simply keeping x close (i.e., in some open U). It is tempting to say «close to x¯ without having to be precisely at x¯», and this works unless the singleton {x¯} is open.

f is then continuous if it is continuous at every point. This is equivalent to the following: For any open V , the inverse image f −1(V ) is open. This is property (ii) in Lindstrøm Proposition 2.3.9, but does not require the space to be metric.

Example / sample problem Let X be the set of bounded piecewise constant functions (with finite number of pieces) defined on [0, 1], and consider the following three topologies R 1 on X: (i) defined by the metric max |x(t) − y(t)|; (ii) defined by the metric: 0 |x(t) − y(t)| dt; (iii) defined by pointwise convergence: a sequence {xn} of functions converges to x if xn(t) tends to x(t) for each t. R T Consider F (x) defined by 0 x(t) dt. That is, F (x) takes x and returns the function f R T defined by f(T ) = 0 x(t) dt. The functions f belong to the space Y of continuous piecewise linear functions (finite number of pieces). We can topologize Y with any of the topologies (i) through (iii) above. With three topologies on X and three topologies on Y, we have nine cases to solve for the following question: Q: Is F (x) continuous wrt. x? (Note: the problems involving pointwise topology are hard, probably too hard for an exam. The 2 × 2 others could maybe be given.)

3.7 Denseness A subset A of a space X is dense in X if any nonempty open set U ⊆ X intersects A, i.e. A ∩ U 6= 0. For example, the rationals form a dense subset Q of the reals R. • Any continuous functions is determined by its values on a dense subset: if a continuous f(q) is known for all q ∈ Q, then for any irrational r we have f(r) = lim f(qn) where {qn} ⊆ Q converges to r.

8 • Also, a continuous function can be constructed by first defining it in a continuous manner on a dense subset. The definition on the complement will follow from taking limits.

Fact: since Q is countable (i.e. it can be covered by a sequence), then any continuous f : R 7→ R might be represented as a sequence. (Actually specifying this representation, is usually grossly inconvenient!)

Example / sample problem Explain why this latter property also holds for any left-continuous f.

4 Measure and integration

Be aware that the Lebesgue integral goes way beyond the integral wrt. the Lebesgue measure, as treated by Lindstrøm. You should:

• be able to work with the basics – measurable space as a space equipped with a σ- algebra, measure space as a measurable space with a measure on it (note that a measurable function or random variable, is a much more important concept than a measurable set);

• understand at an intuitive level, from denseness – why the integral starts with simple functions;

• know that Lebesgue integrals include both (proper) ordinary integrals and sums and expectations with respect to arbitrary probability measures;

• be able to use monotone and dominated convergence (at least the bounded convergence simpliﬁcation) on simple examples, even if the Lebesgue integral is e.g. stated as a sum.

In addition, you should be able to use Tonelli’s theorem to interchange e.g. expectation and sum, over positive functions, and in cases where integrability is not too hard to establish (or given in the problem), do the same with Fubini’s theorem. As the references do not state these, they are given in a simpliﬁed version here:

4.1 Fubini’s and Tonelli’s theorems Let (X, M, µ) and (Y, N , ν) be two σ-ﬁnite measure spaces. Then sufﬁcient for the equality Z h Z i Z h Z i f(x, y) dν(y) dµ(x) = f(x, y) dµ(x) dν(y) X Y Y X

is f is integrable in either the left or right hand side order («Fubini» – i.e., one of the integrals exist and is ﬁnite), or that f is nonnegative («Tonelli»).

Notes:

9 • In order to apply the Fubini version, you might need Tonelli to prove integrability. Suppose that the leftmost integrals represent the «difﬁcult» order of integration, and you want to interchange to the «easy» right hand side order. You can do so for the integral of |f| (since it is nonnegative, Tonelli applies), and then hopefully show that R h R i Y X |f(x, y)| dµ(x) dν(y) is ﬁnite. If so, then Fubini’s theorem applies and you can interchange the order of integration for f itself.

• The full Fubini/Tonelli theorems also state that under the sufﬁcient conditions pre- R sented, the iterated integrals also equal X×Y f(x, y) d(µ ⊗ ν)(x, y), which is the integral wrt. the product measure of µ and ν. However, for the purpose of this course you can disregard this construction (which would have done away with the cumbersome statement of the Fubini case, though.)

Sample problems A few of the problems assigned earlier, were tricky to prove from scratch, and you are free to use the propositions given by Lindstrøm. These problems are somewhat different:

1. Let a ∈ (0, 1) be a constant. Consider the utility function v(c) = 1 − e−c, and let for each t = 0, 1, 2,... , C(t) be uniform over [0, et]. Calculate rigorously the expectation

∞ X E[ a−tv(C(t))] t=0

(You will get a series you cannot calculate explicitely, but without an expectation.) P 2. Let qi be a sequence of positive numbers so that i qi < ∞, and let for each t = 0, 1, 2,... , pt be a probability distribution on t + 1, t + 2,... , with point masses pt(i) for i = t + 1, t + 2,... . Calculate ∞ X lim (qipt(i)). t→∞ i=0

3. Problem 5 (a) from the 2011 sample problem set.

4. Let µ be a set function such that µ(A) = 0 if A is at most countable, and ∞ if A is uncountable. Clearly, µ is null if the underlying space X is finite, so let us assume it is infinite. • What is the minimal sigma-algebra it is defined on, in order to assign ∞ to any uncountable subset of X and 0 to the others? Is µ a measure on this sigma-algebra? • Can µ be a measure on a larger sigma-algebra? Does the underlying space matter, as long as it is infinite? • Not exactly core curriculum, but: whenever µ is a measure, check sigma-finiteness.

The previous problem 3 This problem was intended solved by dominated convergence, but can be done without. It doesn’t even require any more measure theory than establishing the expectation.

10 Let T ∈ N ∪ {∞} and denote by F (x0,T ) the value function of the optimal stopping problem −ρτ 1/3 sup E[e arctan(Xτ )χτ<∞], xt+1 = (xt) + Zt+1, τ≤T where Zt+1 are i.i.d. with P[Zt = 1] = P[Zt = −1] = 1/2, and the interpretation of the χτ<∞ is that if T = ∞ and we choose never to stop (i.e. τ = ∞) then the payoff is zero. 3 Claim: the the infinite-horizon value function F (x0, ∞), exists , and equals lim F (x0,T ). T →∞ To prove this, notice first that arctan is bounded, so the supremum exists whether T is finite or not. Furthermore, increasing T only increases the opportunity set, so F is increasing wrt. T with F (x0, ∞) ≥ limT F (x0,T ). To establish the reverse inequality, it is tempting to take the maximizer for the infinite-horizon problem and truncate it at T . However (see the footnote), there need not be a maximizer. But let > 0 be given, and let τ attain at least F (x0, ∞)−. Let τ,T = min{T, τ}. Since the range of arctan is (−π/2, π/2), the payoff using −ρT 4 τ,T is close to the payoff using τ; at worst it is e π close. As long as T ≥ some T, this is < , and the payoff is ≥ F (x0, ∞)−2. On the other hand, τ,T is admissible in the T -horizon problem, and therefore yields at most F (x0,T ). We have proven: for each > 0, there is a T such that

F (x0, ∞) − 2 ≤ F (x0,T ) (< F (x0, ∞)) ∀ T ≥ T

5 Vector spaces

Reading list here is Lindstrøm chapter 4.5 (for normed spaces and Banach spaces) and 5.8 p d (the L (R ) spaces), in addition to the following. We shall only cover vector spaces over the ﬁeld K = R (i.e., the scalars are real numbers, not complex ones).

5.1 Some norms and normed spaces Lindstrøm defines vector spaces, norms and normed spaces. (Terminology: a Banach space is a normed space which is (Cauchy-) complete.) On the outset, one starts with a space and defines a norm on it. In practice though, one can start with a «too large space», define a suitable functional, then restrict the space so that this is finite. For example, start with the space of sequences of real numbers. Suppose we P 2 1/2 wish to use ( i xi ) as norm; then we restrict ourselves to the subspace for which this is finite, and we have a normed space.

Examples

k×` p • The trace norm on X = R , deﬁned by tr(AA>), is a norm on X (inspecting diago- nal element (i, i) of the matrix product we see that it is the dot product of row #i with P 2 1/2 itself, so the trace norm equals ( ij aij) ).

3Note: the «sup» was written «max» in the earlier version; max is attained in the finite-horizon case, as there are only a finite number of times × states. The max need not be attained in the infinite-horizon case (and I don’t know whether it will be); if the dynamics were xt+1 = xt + Zt, then in the limit, a law-of-iterated-logs argument would lead to the sup being π/2, which is not attained 4 we can compute T = (ln(π/))/ρ, but that is not important

11 ` • The operator norm of a linear operator A from a normed space X (say, R ) to another k normed space Y (say, = R ) is

kAk = sup kAxk kxk≤1

– notice that kxk is the norm in X and kAxk is the norm in Y, while kAk is the norm on the space of those linear functionals for which this is ﬁnite5. For example, in the Eu- k×` clidean case, A will be left-multiplication by some R matrix, which we also denote by A. The operator norm turns out6 to be the square root of the largest eigenvalue of A>A (which is a positive semideﬁnite matrix).

• The set of measures do not form a vector space, since the negative of a measure is not a measure, but taking linear combinations of finite measures (the restricion to finite measures in order to avoid ∞ − ∞), we obtain the vector space of finite signed measures. We can equip the space with the total variation norm Z kµk = sup f dµ. |f|≤1

If P and Q are probability measures, the induced metric becomes (note the «2»!)

d(P,Q) = kP − Qk = 2 sup |P (A) − Q(A)|. A∈F

Sample problems:

• Show that the trace norm is indeed a norm.

• (Hard!) Norm the space of signed ﬁnite measures by the total variation norm. Show R that the operator norm of the functional Tf (µ) = f dµ on the space of ﬁnite signed measures, is indeed supt |f(t)|. You may assume that |f| actually attains a maximum. (This is for simplicity. Otherwise, form a sequence {tn} to approximate supt |f(t)|.)

5.2 The `p sequence spaces and the Lp function spaces, p ∈ [0, ∞]

Let us start briefly with the sequence spaces. Let x = (t1, t2,... ) and y = (u1, u2,... ) be sequences, and think of them as just infinite-dimensional extensions of Maths 2 vectors. The dot product x·y is well-defined as long as the sum converges. For example, if x is a bounded P sequence, then it is sufficient for convergence that i ui converges. But if the convergence P is not absolute – i.e., if i |ui| diverges – then there is a bounded sequence x for which the ∞ dot product diverges (namely, choose ti as the sign of ui). So if we define the space ` of

5if Y = R – whose norm is the absolute value! – then this is called the dual (sometimes «topological dual», sometimes «continuous dual») of X. 6Squaring the problem, we get to max x>(A>A)x subject to x>x = 1. By the Lagrange ﬁrst-order condition, > p (A A)x = λx – i.e., x is an eigenvector. Obviously the eigenvector maximizing (Ax)>Ax is the one for which the eigenvalue is largest (recall from Maths 3 that all eigenvalues of a positive semideﬁnite matrix, are real and ≥ 0, hence «largest» here really means «largest positive», not merely largest absolute value or modulus. You will probably also recall from Mathematics 3 that computing eigenvalues requires solving polynomial equations, and should be kept at low dimension for exam purposes ...)

12 bounded sequences, and `1 as the absolutely summable sequences, then the `1 sequences form bounded linear functionals on `∞, and vice versa. P p1/p More generally, deﬁne the p-norm as kxk∞ := supi |ti| for p = ∞ and as kxkp := i |ti| for p ∈ [1, ∞). Then the space `p consists of those sequences for which the p-norm is ﬁnite, for p ∈ [1, ∞]. It is possible to show that as p → ∞, the p-norm tends to kxk∞ := supi |ti| – however, it will be slightly different for L∞.

For a measure space (X, M, µ), we define as in Lindstrøm, Lp(µ), for p ≥ 1 to be the class 7 p R p of real functions f so that the L norm kfkp := X |f| dµ is finite. In addition we define the infinity norm kfk∞ as the limit of p-norms; it turns out that this norm is the µ-essential supremum of |f|, namely the smallest number which |f| only exceeds on a null set for µ. L∞ is then those functions which have finite ∞-norm. All Lp are complete, i.e. Banach. Notice p p that for all p ∈ [1, ∞], ` = L (N, power set, counting measure).

The Hölder inequality then says that

kfgk1 ≤ kfkp · kgkq

1 1 holds whenever p and q both ∈ [1, ∞] and satisfy p + q = 1 (for probability measures, we can weaken the latter to «≤ 1»). By virtue of the Hölder inequality, we will have that R p q Tg(f) := X |fg| dµ is ﬁnite for f ∈ L , whenever g ∈ L .

On the Itô isometry In the context of Itô integrals, we have covered the multidimensional case, where the functions are vector-valued. In this case, we could just apply the L2 theory component-wise. With the ordinary Euclidean inner product, then v>(t)v(t) is in L2(R) if and only if each component is. Since a vector-valued Brownian motions is just a vector of Brownian motions, then L2(P ) works that way too – and also the dP ⊗ dt integral. We can therefore apply the theory of real-valued functions, and you are not supposed to be concerned about any surprises.

7strictly speaking of equivalence classes of functions equal almost everywhere – you are not supposed to make this point over and over, but you are actually supposed not to discard the p-norm for the reason of it evalu- ating to zero functions which are zero merely almost everywhere!

13 Sample problem set 2012 This problem set takes problems 1–3 from last year’s exam, and continuous-time control was no part of the 2011 curriculum. This is the reason why the set is light on that topic; I tried to keep problems 4–5 not too much exceeding the workload of the remaining part of 2011, so problem 5 is likely a bit too easy.

The font for spaces: Below I applied the same font as in the 2011 exam, where by popular demand something else than old German style was chosen. I might consider that this year as well.

Exam 2011 problem 1 Let β ∈ (0, 1) and N ∈ {0} ∪ N ∪ {∞} be constants, and consider Pτ−1 the optimal stopping problem (where « t=0 » should be interpreted as zero if τ = 0):

τ−1 h X βτ i sup E βtx + x · 1 where x = x + Z with x given, t 1 − β τ τ<∞ t+1 t t+1 0 τ∈{0, ··· ,N} t=0 and where the Zt are independent, and for each t there is a pt ∈ [0, 1] such that Zt = pt or Zt = pt − 1, with respective probabilities Pr[Zt = pt] = 1 − pt and Pr[Zt = pt − 1] = pt. Observe that all Zt have expectation equal to 0.

(a) Assume that N < ∞. Use stochastic dynamic programming to show that at every time t ∈ {0, ··· ,N}, one will be indifferent between stopping or continuation. (Note that it is compulsory to use dynamic programming in this part.)

This shows that any stopping time for which part (a) applies, will yield the same payoff – x0 namely 1−β . In the following parts, you are going to show that this holds also when we allow unbounded and even non-ﬁnite stopping times. Let an arbitrary stopping time τ be given, and deﬁne for each (non-random) n ∈ N the stopping time θn = min{n, τ}. Put

θ −1 h Xn βθn i H = E βtx + x . n t 1 − β θn t=0

x0 (b) Explain why part (a) shows that Hn equals 1−β , for each n ∈ N and each stopping time τ.

τ−1 τ h X t β i lim Hn = E β xt + xτ · 1τ<∞ n→∞ 1 − β t=0

P (a+bt)βt < ∞ a, b Hints: You can use without proof that t∈N for all real numbers . Also, it is probably helpful to consider the sum as an integral in the Lebesgue sense.

i Exam 2011 problem 2 (modiﬁed) Let F : R 7→ [0, 1] be continuous and strictly increasing. (a) Deﬁne

d(x, y) = |F (x) − F (y)| (D)

for all real-valued x, y. Show that (R, d) is a metric space, and that xn → x¯ in (R, d) if and only if xn → x¯ in R with the standard metric |x − y|.

(b) Let X = {−∞} ∪ R ∪ {+∞}. Extend the domain of F by deﬁning F (−∞) = 0 and F (+∞) = 1, and deﬁne d by formula (D), but now valid for all x, y in X. Show that d then becomes a metric on X.

(c) Let (X, d) be the metric space of part (b), and deﬁne the sequence {xn}n∈N by xn = n, all n ∈ N. Show that xn → +∞ in (X, d) if and only if supt∈R F (t) = 1.

Exam 2011 problem 3 Deﬁne the topological space Y as the set of real numbers equipped with the topology generated by taking all sets of the form [a, b) (where −∞ < a < b < ∞), as open in Y. (That is, with these sets as a subbase.) We shall distinguish between topologies by writing convergence «in Y» when referring to this topology, and convergence «in R» when referring to the standard topology.

(a) Show that if xn → x¯ in Y, then xn → x¯ in R.

(b) Show that even if xn → x¯ in R, it does not necessarily follow that the sequence {xn}n∈N converges in Y, and describe – in words if you prefer so – those sequences which converge in Y.

1 4 Let Y be the set of functions f : [0, 1] 7→ R which are C with bounded derivative, and let 0 ν(f) = supt∈[0,1] |f (t)|.

(a) Show that ν is not a norm on Y. n (b) Specify a subspace Z ⊂ Y such that all the functions t are ∈ Z for all n = 1, 2,... , and such that ν is a norm on Z.

5 Let Xt = x+Bt where B is standard Brownian, and let T be ﬁrst time X exits [0, 4]. Solve the optimal stopping problem

sup E[(Xτ − 1)(Xτ − 2)|X0 = x] τ≤T

ii (more info)

topological space (Definition)

"topological space" is owned by djao. [ full author list (2) ] Dense set 1 Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if any point x in X belongs to A or is a limit point of A.[1] Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A - for instance, every real number is either a rational number or has one arbitrarily close to it (see Diophantine approximation). Formally, a subset A of a topological space X is dense in X if for any point x in X, any neighborhood of x contains at least one point from A. Equivalently, A is dense in X if and only if the only closed subset of X containing A is X itself. This can also be expressed by saying that the closure of A is X, or that the interior of the complement of A is empty. The density of a topological space X is the least cardinality of a dense subset of X.

Density in metric spaces An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points),

Then A is dense in X if

Note that . If is a sequence of dense open sets in a complete metric space, X, then is also dense in X. This fact is one of the equivalent forms of the Baire category theorem.

Examples The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets. By the Weierstrass approximation theorem, any given complex-valued continuous function defined on a closed interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. In other words, the polynomial functions are dense in the space C[a,b] of continuous complex-valued functions on the interval [a,b], equipped with the supremum norm. Every metric space is dense in its completion.

Properties Every topological space is dense in itself. For a set X equipped with the discrete topology the whole space is the only dense set. Every non-empty subset of a set X equipped with the trivial topology is dense, and every topology for which every non-empty subset is dense must be trivial. Denseness is transitive: Given three subsets A, B and C of a topological space X with A ⊆ B ⊆ C such that A is dense in B and B is dense in C (in the respective subspace topology) then A is also dense in C. The image of a dense subset under a surjective continuous function is again dense. The density of a topological space is a topological invariant. A topological space with a connected dense subset is necessarily connected itself. Dense set 2

Continuous functions into Hausdorff spaces are determined by their values on dense subsets: if two continuous functions f, g : X → Y into a Hausdorff space Y agree on a dense subset of X then they agree on all of X.

Related notions A point x of a subset A of a topological space X is called a limit point of A (in X) if every neighbourhood of x also contains a point of A other than x itself, and an isolated point of A otherwise. A subset without isolated points is said to be dense-in-itself. A subset A of a topological space X is called nowhere dense (in X) if there is no neighborhood in X on which A is dense. Equivalently, a subset of a topological space is nowhere dense if and only if the interior of its closure is empty. The interior of the complement of a nowhere dense set is always dense. The complement of a closed nowhere dense set is a dense open set. A topological space with a countable dense subset is called separable. A topological space is a Baire space if and only if the intersection of countably many dense open sets is always dense. A topological space is called resolvable if it is the union of two disjoint dense subsets. More generally, a topological space is called κ-resolvable if it contains κ pairwise disjoint dense sets. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. A linear operator between topological vector spaces X and Y is said to be densely defined if its domain is a dense subset of X and if its range is contained within Y. See also continuous linear extension. A topological space X is hyperconnected if and only if every nonempty open set is dense in X. A topological space is submaximal if and only if every dense subset is open.

References

Notes [1] Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 048668735X

General references • Nicolas Bourbaki (1989) [1971]. General Topology, Chapters 1–4. Elements of Mathematics. Springer-Verlag. ISBN 3-540-64241-2. • Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR507446 Article Sources and Contributors 3 Article Sources and Contributors

Dense set Source: http://en.wikipedia.org/w/index.php?oldid=456661398 Contributors: A19grey, Aleph4, Archelon, Austinmohr, Banus, Bdmy, Bluestarlight37, Calle, Can't sleep, clown will eat me, CiaPan, Cuaxdon, Denis.arnaud, Dino, ELLusKa 86, Erzbischof, Fly by Night, Giftlite, Graham87, Haemo, Headbomb, Isnow, Kae1is, Macrakis, Maksim-e, MathMartin, Michael Hardy, anonymous edits דניאל ב., דניאל צבי, Mks004, Nguyen Thanh Quang, Oleg Alexandrov, Physicistjedi, Poulpy, Tobias Bergemann, Utkwes, Vundicind, 25 License

Creative Commons Attribution-Share Alike 3.0 Unported //creativecommons.org/licenses/by-sa/3.0/ Measure (mathematics) 1 Measure (mathematics)

In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space Rn. For instance, the Lebesgue measure of the interval [0, 1] in the real numbers is its length in the everyday sense of the word, specifically 1.

To qualify as a measure (see Definition below), a function that assigns a non-negative real number or +∞ to a set's subsets must satisfy a few conditions. One important condition is countable additivity. This condition states that the size of the union of a sequence of disjoint subsets is equal to the sum of the sizes of the subsets. However, it is in general impossible to associate a consistent size to each subset of a given set and also satisfy the other axioms of a measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the subsets on which the measure is to be defined are called measurable and they are required to form a σ-algebra, meaning that Informally, a measure has the property of being monotone in the sense that if A is a subset of B, unions, intersections and complements of sequences of measurable the measure of A is less than or equal to the subsets are measurable. Non-measurable sets in a Euclidean space, on measure of B. Furthermore, the measure of the which the Lebesgue measure cannot be defined consistently, are empty set is required to be 0. necessarily complicated, in the sense of being badly mixed up with their complements; indeed, their existence is a non-trivial consequence of the axiom of choice.

Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.

Definition Let Σ be a σ-algebra over a set X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties: • Non-negativity: for all • Countable additivity (or σ-additivity): For all countable collections of pairwise disjoint sets in Σ:

• Null empty set: Measure (mathematics) 2

One may require that at least one set E has finite measure. Then the null set automatically has measure zero because

of countable additivity, because and is

finite if and only if the empty set has measure zero. The pair (X, Σ) is called a measurable space, the members of Σ are called measurable sets, and the triple (X, Σ, μ) is called a measure space. If only the second and third conditions of the definition of measure above are met, and μ takes on at most one of the values ±∞, then μ is called a signed measure. A probability measure is a measure with total measure one (i.e., μ(X) = 1); a probability space is a measure space with a probability measure. For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures. Radon measures have an alternative definition in terms of linear functionals on the locally convex space of continuous functions with compact support. This approach is taken by Bourbaki (2004) and a number of other authors. For more details see Radon measure.

Properties Several further properties can be derived from the definition of a countably additive measure.

Monotonicity A measure μ is monotonic: If E and E are measurable sets with E ⊆ E then 1 2 1 2

Measures of infinite unions of measurable sets A measure μ is countably subadditive: If E , E , E , … is a countable sequence of sets in Σ, not necessarily disjoint, 1 2 3 then

A measure μ is continuous from below: If E , E , E , … are measurable sets and E is a subset of E for all n, then 1 2 3 n n + 1 the union of the sets E is measurable, and n Measure (mathematics) 3

Measures of infinite intersections of measurable sets A measure μ is continuous from above: If E , E , E , … are measurable sets and E is a subset of E for all n, then 1 2 3 n + 1 n the intersection of the sets E is measurable; furthermore, if at least one of the E has finite measure, then n n

This property is false without the assumption that at least one of the E has finite measure. For instance, for each n ∈ n N, let

which all have infinite Lebesgue measure, but the intersection is empty.

Sigma-finite measures A measure space (X, Σ, μ) is called finite if μ(X) is a finite real number (rather than ∞). It is called σ-finite if X can be decomposed into a countable union of measurable sets of finite measure. A set in a measure space has σ-finite measure if it is a countable union of sets with finite measure. For example, the real numbers with the standard Lebesgue measure are σ-finite but not finite. Consider the closed intervals [k,k+1] for all integers k; there are countably many such intervals, each has measure 1, and their union is the entire real line. Alternatively, consider the real numbers with the counting measure, which assigns to each finite set of reals the number of points in the set. This measure space is not σ-finite, because every set with finite measure contains only finitely many points, and it would take uncountably many such sets to cover the entire real line. The σ-finite measure spaces have some very convenient properties; σ-finiteness can be compared in this respect to the Lindelöf property of topological spaces. They can be also thought of as a vague generalization of the idea that a measure space may have 'uncountable measure'.

Completeness A measurable set X is called a null set if μ(X)=0. A subset of a null set is called a negligible set. A negligible set need not be measurable, but every measurable negligible set is automatically a null set. A measure is called complete if every negligible set is measurable. A measure can be extended to a complete one by considering the σ-algebra of subsets Y which differ by a negligible set from a measurable set X, that is, such that the symmetric difference of X and Y is contained in a null set. One defines μ(Y) to equal μ(X).

Additivity Measures are required to be countably additive. However, the condition can be strengthened as follows. For any set I and any set of nonnegative r , define: i

A measure on is -additive if for any and any family , the following hold:

Note that the second condition is equivalent to the statement that the ideal of null sets is -complete. Measure (mathematics) 4

Examples Some important measures are listed here. • The counting measure is defined by μ(S) = number of elements in S. • The Lebesgue measure on R is a complete translation-invariant measure on a σ-algebra containing the intervals in R such that μ([0,1]) = 1; and every other measure with these properties extends Lebesgue measure. • Circular angle measure is invariant under rotation, and hyperbolic angle measure is invariant under squeeze mapping. • The Haar measure for a locally compact topological group is a generalization of the Lebesgue measure (and also of counting measure and circular angle measure) and has similar uniqueness properties. • The Hausdorff measure is a generalization of the Lebesgue measure to sets with non-integer dimension, in particular, fractal sets. • Every probability space gives rise to a measure which takes the value 1 on the whole space (and therefore takes all its values in the unit interval [0,1]). Such a measure is called a probability measure. See probability axioms. • The Dirac measure δ (cf. Dirac delta function) is given by δ (S) = χ (a), where χ is the characteristic function of a a S S S. The measure of a set is 1 if it contains the point a and 0 otherwise. Other 'named' measures used in various theories include: Borel measure, Jordan measure, ergodic measure, Euler measure, Gaussian measure, Baire measure, Radon measure and Young measure. In physics an example of a measure is spatial distribution of mass (see e.g., gravity potential), or another non-negative extensive property, conserved (see conservation law for a list of these) or not. Negative values lead to signed measures, see "generalizations" below. Liouville measure, known also as the natural volume form on a symplectic manifold, is useful in classical statistical and Hamiltonian mechanics. Gibbs measure is widely used in statistical mechanics, often under the name canonical ensemble.

Non-measurable sets If the axiom of choice is assumed to be true, not all subsets of Euclidean space are Lebesgue measurable; examples of such sets include the Vitali set, and the non-measurable sets postulated by the Hausdorff paradox and the Banach–Tarski paradox.

Generalizations For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instance, a countably additive set function with values in the (signed) real numbers is called a signed measure, while such a function with values in the complex numbers is called a complex measure. Measures that take values in Banach spaces have been studied extensively. A measure that takes values in the set of self-adjoint projections on a Hilbert space is called a projection-valued measure; these are used in functional analysis for the spectral theorem. When it is necessary to distinguish the usual measures which take non-negative values from generalizations, the term positive measure is used. Positive measures are closed under conical combination but not general linear combination, while signed measures are the linear closure of positive measures. Another generalization is the finitely additive measure, which are sometimes called contents. This is the same as a measure except that instead of requiring countable additivity we require only finite additivity. Historically, this definition was used first, but proved to be not so useful. It turns out that in general, finitely additive measures are connected with notions such as Banach limits, the dual of L∞ and the Stone–Čech compactification. All these are linked in one way or another to the axiom of choice. A charge is a generalization in both directions: it is a finitely additive, signed measure. Measure (mathematics) 5

References • Robert G. Bartle (1995) The Elements of Integration and Lebesgue Measure, Wiley Interscience. • Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1 Chapter III. • R. M. Dudley, 2002. Real Analysis and Probability. Cambridge University Press. • Folland, Gerald B. (1999), Real Analysis: Modern Techniques and Their Applications, John Wiley and Sons, ISBN 0-471-317160-0 Second edition. • D. H. Fremlin, 2000. Measure Theory [1]. Torres Fremlin. • Paul Halmos, 1950. Measure theory. Van Nostrand and Co. • R. Duncan Luce and Louis Narens (1987). "measurement, theory of," The New Palgrave: A Dictionary of Economics, v. 3, pp. 428–32. • M. E. Munroe, 1953. Introduction to Measure and Integration. Addison Wesley. • K. P. S. Bhaskara Rao and M. Bhaskara Rao (1983), Theory of Charges: A Study of Finitely Additive Measures, London: Academic Press, pp. x + 315, ISBN 0-1209-5780-9 • Shilov, G. E., and Gurevich, B. L., 1978. Integral, Measure, and Derivative: A Unified Approach, Richard A. Silverman, trans. Dover Publications. ISBN 0-486-63519-8. Emphasizes the Daniell integral. • Jech, Thomas (2003), Set Theory: The Third Millennium Edition, Revised and Expanded, Springer Verlag, ISBN 3-540-44085-2

External links • Tutorial: Measure Theory for Dummies [2] • Yet another, yet very reader-friendly, introduction to the measure theory (with application to quantitative finance in mind) [3]

References

[1] http:/ / www. essex. ac. uk/ maths/ people/ fremlin/ mt. htm

[2] http:/ / www. ee. washington. edu/ techsite/ papers/ documents/ UWEETR-2006-0008. pdf

[3] http:/ / www. yetanotherquant. de Article Sources and Contributors 6 Article Sources and Contributors

Measure (mathematics) Source: http://en.wikipedia.org/w/index.php?oldid=461329451 Contributors: 16@r, 3mta3, 7&6=thirteen, ABCD, Aetheling, AiusEpsi, Akulo, Alansohn, Albmont, AleHitch, Andre Engels, Arvinder.virk, Ashigabou, Avaya1, AxelBoldt, Baaaaaaar, Bdmy, Beaumont, BenFrantzDale, Benandorsqueaks, Bgpaulus, Boobahmad101, Boplin, Brad7777, Brian Tvedt, BrianS36, CRGreathouse, CSTAR, Caesura, Cdamama, Charles Matthews, Charvest, Conversion script, Crasshopper, DVdm, Danielbojczuk, Daniele.tampieri, Dark Charles, Dave Ordinary, DealPete, Digby Tantrum, Dino, Discospinster, Dowjgyta, Dpv, Dysprosia, EIFY, Edokter, Elwikipedista, Empty Buffer, Everyking, Fibonacci, Finanzmaster, Finell, Foxjwill, GIrving, Gabbe, Gadykozma, Gaius Cornelius, Gandalf61, Gar37bic, Gauge, Geevee, Geometry guy, Giftlite, Gilliam, Googl, Harriv, Henning Makholm, Hesam7, Irvin83, Isnow, Iwnbap, Jackzhp, Jay Gatsby, Jheald, Jorgen W, Joriki, Juliancolton, Jóna Þórunn, Keenanpepper, Kiefer.Wolfowitz, Lambiam, Le Docteur, Lethe, Levineps, Linas, Loisel, Loren Rosen, Lupin, MABadger, MER-C, Manop, MarSch, Markjoseph125, Masterpiece2000, Mat cross, MathKnight, MathMartin, Matthew Auger, Mebden, Melcombe, Michael Hardy, Michael P. Barnett, Miguel, Mike Segal, Mimihitam, Mousomer, MrRage, Msh210, Myasuda, Nbarth, Nicoguaro, Obradovic Goran, Ocsenave, Oleg Alexandrov, OverlordQ, Paolo.dL, Patrick, Paul August, PaulTanenbaum, Pdenapo, PhotoBox, Pmanderson, Point-set topologist, Pokus9999, Prumpf, Ptrf, RMcGuigan, Rat144, RayAYang, Revolver, Rgdboer, Richard L. Peterson, Rktect, S2000magician, Salgueiro, Salix alba, SchfiftyThree, Semistablesystem, Stca74, Sullivan.t.j, Sverdrup, Sławomir Biały, TakuyaMurata, Takwan, Tcnuk, The Infidel, The Thing That Should Not Be, Thehotelambush, Thomasmeeks, Tobias Bergemann, Toby, Toby Bartels, Tosha, Tsirel, Turms, Uranographer, Vivacissamamente, Weialawaga, Xantharius, Zero sharp, Zhangkai Jason Cheng, Zundark, Zvika, 145 anonymous edits Image Sources, Licenses and Contributors

Image:Measure illustration.png Source: http://en.wikipedia.org/w/index.php?title=File:Measure_illustration.png License: Public Domain Contributors: Oleg Alexandrov License

Creative Commons Attribution-Share Alike 3.0 Unported //creativecommons.org/licenses/by-sa/3.0/ Measurable function 1 Measurable function

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. This definition can be deceptively simple, however, as special care must be taken regarding the -algebras involved. In particular, when a function is said to be Lebesgue measurable what is actually meant is that is a measurable function—that is, the domain and range represent different -algebras on the same underlying set (here is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on ). 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 generated by its open subsets 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. 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.

Formal definition Let and be measurable spaces, meaning that and are sets equipped with respective sigma algebras and . A function

is said to be measurable if for every . The notion of measurability depends on the sigma algebras and . To emphasize this dependency, if is a measurable function, we will write

Special measurable functions • If and are Borel spaces, a measurable function 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 , it is called a Borel section. • A Lebesgue measurable function is a measurable function , where is the sigma algebra of Lebesgue measurable sets, and is the Borel algebra on the complex numbers . Lebesgue measurable functions are of interest in mathematical analysis because they can be integrated. • Random variables are by definition measurable functions defined on sample spaces. Measurable function 2

Properties of measurable functions • The sum and product of two complex-valued measurable functions are measurable.[2] So is the quotient, so long as there is no division by zero.[1] • The composition of measurable functions is measurable; i.e., if and are measurable functions, then so is .[1] 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] [3] • The pointwise limit of a sequence of measurable functions is measurable; 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.[4] )

Non-measurable functions Real-valued functions encountered in applications tend to be measurable; however, it is not difficult to find non-measurable functions. • So long as there are non-measurable sets in a measure space, there are non-measurable functions from that space. If is some measurable space and is a non-measurable set, i.e. if , then the indicator function is non-measurable (where is equipped with the Borel algebra as usual), since the preimage of the measurable set is the non-measurable set . Here is given by

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

Notes [1] Strichartz, Robert (2000). The Way of Analysis. Jones and Bartlett. ISBN 0-7637-1497-6. [2] Folland, Gerald B. (1999). Real Analysis: Modern Techniques and their Applications. Wiley. ISBN 0471317160. [3] Royden, H. L. (1988). Real Analysis. Prentice Hall. ISBN 0-02-404151-3. [4] Dudley, R. M. (2002). Real Analysis and Probability (2 ed.). Cambridge University Press. ISBN 0521007542. Article Sources and Contributors 3 Article Sources and Contributors

Measurable function Source: http://en.wikipedia.org/w/index.php?oldid=446563435 Contributors: AxelBoldt, Babcockd, Bdmy, CSTAR, Calle, Charles Matthews, Chungc, CàlculIntegral, Dan131m, Dingenis, Dysprosia, Fibonacci, Fropuff, Gala.martin, GeeJo, GiM, Giftlite, Ht686rg90, Imran, Jahredtobin, Jka02, Linas, Loewepeter, Mandarax, Maneesh, MarSch, Miguel, Mike Segal, Msh210, OdedSchramm, Oleg Alexandrov, PV=nRT, Parodi, Pascal.Tesson, Paul August, Richard L. Peterson, Rinconsoleao, Ron asquith, Rs2, Salgueiro, Salix alba, Silverfish, Sławomir Biały, Takwan, Tosha, Vivacissamamente, Walterfm, Weialawaga, Zundark, Zvika, 37 anonymous edits License

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