Chapter 5 Product Measures

Chapter 5 Product Measures Lebesgue measure on R generalizes the notion of the length of an interval. In this chapter, we see how two-dimensional Lebesgue measure on R2 generalizes the notion of the area of a rectangle. More generally, we construct new measures that are the products of two measures. Once these new measures have been constructed, the question arises of how to compute integrals with respect to these new measures. Beautiful theorems proved in the first decade of the twentieth century allow us to compute integrals with respect to product measures as iterated integrals involving the two measures that produced the product. Furthermore, we will see that under reasonable conditions we can switch the order of an iterated integral. Main building of Scuola Normale Superiore di Pisa, the university in Pisa, Italy, where Guido Fubini (1879–1943) received his PhD in 1900. In 1907 Fubini proved that under reasonable conditions, an integral with respect to a product measure can be computed as an iterated integral and that the order of integration can be switched. Leonida Tonelli (1885–1943) also taught for many years in Pisa; he also proved a crucial theorem about interchanging the order of integration in an iterated integral. CC-BY-SA Lucarelli © Sheldon Axler 2020 S. Axler, Measure, Integration & Real Analysis, Graduate Texts 116 in Mathematics 282, https://doi.org/10.1007/978-3-030-33143-6_5 Section 5A Products of Measure Spaces 117 5A Products of Measure Spaces Products of s-Algebras Our first step in constructing product measures is to construct the product of two s-algebras. We begin with the following definition. 5.1 Definition rectangle Suppose X and Y are sets. A rectangle in X Y is a set of the form A B, × × where A X and B Y. ⊂ ⊂ Keep the figure shown here in mind when thinking of a rectangle in the sense defined above. However, remember that A and B need not be intervals as shown in the figure. Indeed, the concept of an interval makes no sense in the generality of arbitrary sets. Now we can define the product of two s-algebras. 5.2 Definition product of two s-algebras; S ⊗ T Suppose (X, ) and (Y, ) are measurable spaces. Then S T the product is defined to be the smallest s-algebra on X Y that • contains S ⊗ T × A B : A , B ; f × 2 S 2 T g a measurable rectangle in is a set of the form A B, where A • S ⊗ T × 2 S and B . 2 T Using the terminology introduced in The notation is not used the second bullet point above, we can say because andS × Tare sets (of sets), that is the smallest s-algebra con- and thusS the notationT tainingS ⊗ all T the measurable rectangles in already is defined to meanS × Tthe set of . Exercise 1 in this section asks all ordered pairs of the form (A, B), youS ⊗ to T show that the measurable rectan- where A and B . gles in are the only rectangles in 2 S 2 T S ⊗ T X Y that are in . ×The notion of crossS ⊗ T sections plays a crucial role in our development of product measures. First, we define cross sections of sets, and then we define cross sections of functions. 118 Chapter 5 Product Measures b 5.3 Definition cross sections of sets; [E]a and [E] Suppose X and Y are sets and E X Y. Then for a X and b Y, the cross b ⊂ × 2 2 sections [E]a and [E] are defined by [E] = y Y : (a, y) E and [E]b = x X : (x, b) E . a f 2 2 g f 2 2 g 5.4 Example cross sections of a subset of X Y × 5.5 Example cross sections of rectangles Suppose X and Y are sets and A X and B Y. If a X and b Y, then ⊂ ⊂ 2 2 B if a A, b A if b B, [A B]a = 2 and [A B] = 2 × Æ if a / A × Æ if b / B, ( 2 ( 2 as you should verify. The next result shows that cross sections preserve measurability. 5.6 cross sections of measurable sets are measurable Suppose is a s-algebra on X and is a s-algebra on Y. If E , then S T 2 S ⊗ T [E] for every a X and [E]b for every b Y. a 2 T 2 2 S 2 Proof Let denote the collection of subsets E of X Y for which the conclusion E × of this result holds. Then A B for all A and all B (by Example 5.5). The collection is closed× under2 E complementation2 S and countable2 T unions because E [(X Y) E] = Y [E] × n a n a and [E E ] = [E ] [E ] 1 [ 2 [··· a 1 a [ 2 a [··· for all subsets E, E , E ,... of X Y and all a X, as you should verify, with 1 2 × 2 similar statements holding for cross sections with respect to all b Y. Because is a s-algebra containing all the measurable rectangles2 in , we conclude thatE contains . S ⊗ T E S ⊗ T Section 5A Products of Measure Spaces 119 Now we define cross sections of functions. b 5.7 Definition cross sections of functions; [ f ]a and [ f ] Suppose X and Y are sets and f : X Y R is a function. Then for a X and × ! b 2 b Y, the cross section functions [ f ]a : Y R and [ f ] : X R are defined by2 ! ! [ f ] (y) = f (a, y) for y Y and [ f ]b(x) = f (x, b) for x X. a 2 2 5.8 Example cross sections Suppose f : R R R is defined by f (x, y) = 5x2 + y3. Then • × ! 3 3 2 [ f ]2(y) = 20 + y and [ f ] (x) = 5x + 27 for all y R and all x R, as you should verify. 2 2 Suppose X and Y are sets and A X and B Y. If a X and b Y, then • ⊂ ⊂ 2 2 [ ] = ( ) [ ]b = ( ) cA B a cA a cB and cA B cB b cA , × × as you should verify. The next result shows that cross sections preserve measurability, this time in the context of functions rather than sets. 5.9 cross sections of measurable functions are measurable Suppose is a s-algebra on X and is a s-algebra on Y. Suppose S T f : X Y R is an -measurable function. Then × ! S ⊗ T [ f ] is a -measurable function on Y for every a X a T 2 and [ f ]b is an -measurable function on X for every b Y. S 2 Proof Suppose D is a Borel subset of R and a X. If y Y, then 2 2 1 y ([ f ] )− (D) [ f ] (y) D 2 a () a 2 f (a, y) D () 2 1 (a, y) f − (D) () 2 1 y [ f − (D)] . () 2 a Thus 1 1 ([ f ]a)− (D) = [ f − (D)]a. Because f is an -measurable function, f 1(D) . Thus the equation S ⊗ T − 2 S ⊗ T above and 5.6 imply that ([ f ] ) 1(D) . Hence [ f ] is a -measurable function. a − 2 T a T The same ideas show that [ f ]b is an -measurable function for every b Y. S 2 120 Chapter 5 Product Measures Monotone Class Theorem The following standard two-step technique often works to prove that every set in a s-algebra has a certain property: 1. show that every set in a collection of sets that generates the s-algebra has the property; 2. show that the collection of sets that has the property is a s-algebra. For example, the proof of 5.6 used the technique above—first we showed that every measurable rectangle in has the desired property, then we showed that the collection of sets that hasS the ⊗ desired T property is a s-algebra (this completed the proof because is the smallest s-algebra containing the measurable rectangles). The techniqueS ⊗ T outlined above should be used when possible. However, in some situations there seems to be no reasonable way to verify that the collection of sets with the desired property is a s-algebra. We will encounter this situation in the next subsection. To deal with it, we need to introduce another technique that involves what are called monotone classes. The following definition will be used in our main theorem about monotone classes. 5.10 Definition algebra Suppose W is a set and is a set of subsets of W. Then is called an algebra A A on W if the following three conditions are satisfied: Æ ; • 2 A if E , then W E ; • 2 A n 2 A if E and F are elements of , then E F . • A [ 2 A Thus an algebra is closed under complementation and under finite unions; a s-algebra is closed under complementation and countable unions. 5.11 Example collection of finite unions of intervals is an algebra Suppose is the collection of all finite unions of intervals of R. Here we are in- cluding all intervals—openA intervals, closed intervals, bounded intervals, unbounded intervals, sets consisting of only a single point, and intervals that are neither open nor closed because they contain one endpoint but not the other endpoint. Clearly is closed under finite unions. You should also verify that is closed under complementation.A Thus is an algebra on R. A A 5.12 Example collection of countable unions of intervals is not an algebra Suppose is the collection of all countable unions of intervals of R.

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