Measure Theory
JesusFernandez-Villaverde University of Pennsylvania
1 Why Bother with Measure Theory?
Kolmogorov (1933).
Foundation of modern probability.
Deals easily with: 1. Continuous versus discrete probabilities. Also mixed probabilities.
2. Univariate versus multivariate.
3. Independence.
4. Convergence.
2 Introduction to Measure Theory
Measure theory is an important eld for economists.
We cannot do in a lecture what it will take us (at least) a whole semester.
Three sources: 1. Read chapters 7 and 8 in SLP.
2. Excellent reference: A User's Guide to Measure Theoretic Proba- bility, by David Pollard.
3. Take math classes!!!!!!!!!!!!!! 3 Algebra
Let S be a set and let be a family of subsets of S. is a algebra S S if
1. ;S . ; 2 S 2. A Ac = S A . 2 S ) n 2 S
3. An , n = 1; 2; :::; An . 2 S )[n1=1 2 S
(S; ): measurable space. S
A : measurable set. 2 S
4 Borel Algebra
De ne a collection of subsets of S: A
algebra generated by : the intersection of all algebra contain- A ing is a algebra. A
algebra generated by is the smallest algebra containing . A A
Example: let be the collection of all open balls (or rectangles) of Rl B (or a restriction of):
Borel algebra: the algebra generated by . B
Borel set: any set in . B 5 Measures
Let (S; ) be a measurable space. S
Measure: an extended real-valued function : R such that: S! 1 1. ( ) = 0. ; 2. (A) 0; A . 8 2 S
3. If An is a countable, disjoint sequence of subsets in , then f gn1=1 S An = (An). [n1=1 n1=1 P If (S) < , then is nite. 1 (S; ; ): measurable space. S 6 Probability Measures
Probability measure: such that (S) = 1.
Probability space: (S; ; ) where is a probability measure. S
Event: each A . 2 S
Probability of an event: (A).
B (S; ): space all probability measures on (S; ) : S S
7 Almost Everywhere
Given (S; ; ), a proposition holds almost everywhere ( a.e.), if S a set A with (A) = 0, such that the proposition holds on Ac: 9 2 S
If is a probability measure, we often use the phrase almost surely (a.s.) instead of almost everywhere.
8 Completion
Let (S; ; ) be a measure space. S
De ne the family of subsets of any set with measure zero: = C S : C A for some A with (A) = 0 C f 2 S g
Completion of is the family : S S0 = B S : B = (B C ) C , B , C ;C S0 0 0 [ 1 n 2 2 S 1 2 2 C n o
(): completion of with respect to measure : S0 S 9 Universal Algebra
= B(S; ) 0 () : U \ 2 S S
Note: 1. is a algebra: U 2. . B U
Universally measurable space is a measurable space with its universal algebra.
Universal algebras avoid a problem of Borel algebras: projection of Borel sets are not necessarily measurable with respect to . B 10 Measurable Function
Measurable function into R: given a measurable space (S; ), a real- S valued function f : S R is measurable with respect to (or measurable) if ! S S s : f (s) a ; a R f 2 S g 2 S 8 2
Measurable function into a measurable space: given two measurable spaces (S; ) and (T; ), the function f : S T is measurable if: S T ! s : f (s) A ; A f 2 S 2 g 2 S 8 2 T
If we set (T; ) = (R; ), the second de nition nests the rst. T B
Random variable: a measurable function in a probability space. 11 Measurable Selection
Measurable selection: given two measurable spaces (S; ) and (T; ) S T and a correspondence of S into T , the function h : S T is a ! measurable selection from is h is measurable and:
h (s) (s), s 2 8 2 S
Measurable Selection Theorem: Let S Rl and T Rm and S and be their universal algebras. Let :S T be a (nonempty) T ! compact-valued and u.h.c. correspondence. Then, a measurable 9 selection from .
12 Measurable Simple Functions
M (S; ): space of measurable, extended real-valued functions on S. S
M+ (S; ): subset of nonnegative functions. S
Measurable simple function: n (s) = a (s) i Ai iX=1
Importance: for any measurable function f, such that (s) 9 f ng n ! f pointwise.
13 Integrals
Integral of with respect to : n (s) (ds) = ai (Ai) S Z iX=1 Integral of f M+ (S; ) with respect to : 2 S f (s) (ds) = sup (s) (ds) ZS (s) M+(S; ) ZS 2 S such that 0 f: Integral of f M+ (S; ) over A with respect to : 2 S
f (s) (ds) = f (s) A (s) (ds) ZA ZS
14 Positive and Negative Parts
We de ne the previous results with positive functions.
How do we extend to the general case?
f +: positive part of a function + f (s) if f (s) 0 f (s) = ( 0 if f (s) < 0
f : negative part of a function f (s) if f (s) 0 f (s) = ( 0 if f (s) > 0
15 Integrability
Let (S; ; ) be a measure space and let f be measurable, real-valued S + function on S. If f and f both have nite integrals with respect to , then f is integrable and the integral is given by
+ fd = f d f d Z Z Z
If A , the integral of f over A with respect to : 2 S + fd = f d f d ZA ZA ZA
16 Transition Functions
Transition function: given a measurable space (Z; ), a function Q : Z Z [0; 1] such that: Z ! 1. For z Z, Q (z; ) is a probability measure on (Z; ). 8 2 Z 2. For A , Q ( ;A) is measurable. 8 2 Z Z
Zt; t = (Z ::: Z; ::: )(t times). Z Z Z t Then, for any rectangle B = A ::: At ; de ne: 1 2 Z t (z0;B) = ::: Q (zt 1; dzt) Q (zt 2; dzt 1) :::Q (z0; dz1) A1 At 1 At Z Z Z
17 Two Operators
For any measurable function f, de ne: Z (T f)(z) = f z Q z; dz , s 0 0 8 2 S Z Interpretation: expected value of f next period.
For any probability measure on (Z; ), de ne: Z
(T )(A) = Q (z; A) (dz), A Z 8 2 Z Interpretation: probability that the state will be in A next period.
18 Basic Properties
T maps the space of bounded -measurable functions, B (Z; ) ; into Z Z itself.
T maps the space of probability measures on (Z; ) ; (Z; ) ; into Z Z itself.
T and T are adjoint operators: (T f)(z) (dz) = f z (T ) dz , (Z; ) 0 0 8 2 Z Z Z for any function f B (Z; ). 2 Z
19 Two Properties
A transition function Q on (Z; ) has the Feller property if the asso- Z ciated operator T maps the space of bounded continuous function on Z into itself.
A transition function Q on (Z; ) is monotone if for every nondecreas- Z ing function f, T f is also non-decreasing.
20 Consequences of our Two Properties
If Z Rl is compact and Q has the Feller property, then a probability 9 measure that is invariant under Q:
= (T )(A) = Q (z; A) (dz) Z
Weak convergence: a sequence n converges weakly to (n ) if f g ) lim fd = fd, f C (S) n n !1 Z Z 8 2
If Q is monotone, has the Feller property, and there is enough \mixing" in the distribution, there is a unique invariant probability measure , and T n for (Z; ). 0 ) 8 0 2 Z
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