Product Measure and Fubini's Theorem

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Lecture 9 Fall 2018 PRODUCT MEASURE AND FUBINI’S THEOREM Contents 1. Product measure 2. Fubini’s theorem In elementary math and calculus, we often interchange the order of summation and integration. The discussion here is concerned with conditions under which this is legitimate. 1 PRODUCT MEASURE Consider two probabilistic experiments with probability spaces ( Ω1 , F1 , P1 ) and (Ω2 , F2 , P2 ), respectively. We are interested in forming a probabilistic model of a “joint experiment” in which the original two experiments are carried out independently. 1.1 The sample space of the joint experiment If the first experiment has an outcome !1 , and the second has an outcome !2 , then the outcome of the joint experiment is the pair (!1 , !2 ). This leads us to define a new sample space Ω = Ω 1 × Ω2 . 1.2 The ˙-algebra of the joint experiment Next, we need a ˙-algebra on Ω . If A1 ∈ F1 , we certainly want to be able to talk about the event {!1 ∈ A1 } and its probability. In terms of the joint experiment, this would be the same as the event A1 × Ω 1 = {(!1 , !2 ) | !1 ∈ A1 , !2 ∈ Ω2 }. 1 Thus, we would like our ˙-algebra on Ω to include all sets of the form A1 × Ω2 , (with A1 ∈ F1 ) and by symmetry, all sets of the form Ω1 × A2 (with (A2 ∈ F2 ). This leads us to the following definition. Definition 1. We define F1 × F2 as the smallest ˙ -algebra of subsets of Ω1 × Ω2 that contains all sets of the form A1 × Ω2 and Ω1 × A2 , where A1 ∈ F1 and A2 ∈ F2 . Note that the notation F1 ×F2 is misleading: this is not the Cartesian product of F1 and F2 ! Since ˙-fields are closed under intersection, we observe that if Ai ∈ Fi , then A1 × A2 = (A1 × Ω2 ) ∩ (Ω1 ∩ A2 ) ∈ F1 × F2 . It turns out (and is not hard to show) that F1 × F2 can also be defined as the smallest ˙-algebra containing all sets of the form A1 × A2 , where Ai ∈ Fi . Alternatively, suppose F1 and F2 are generated by algebras F0,1 , F0,2 . That is Fi = ˙(F0,i ), i = 1, 2. Then F1 × F2 is also the smallest ˙-algebra containing all sets of the form A1 × A2 , where Ai ∈ F0,i . In the sequel, we will talk about g : Ω 1 × Ω 2 → R – measurable functions with respect to F1 × F2 . Recall, this means that for any Borel set B ⊂ R, the set {(!1 , !2 ) | g(!1 , !2 ) ∈ B} belongs to the ˙-algebra F1 × F2 . As a practical matter, it is enough to verify that for any scalar c, the set {(!1 , !2 ) | g(!1 , !2 ) ≤ c} is measurable. Other than using this definition directly, how else can we verify that such a function g is measurable? The basic tools at hand are the following: (a) continuous functions from R2 to R are measurable; (b) indicator functions of measurable sets are measurable; (c) combining measurable functions in the usual ways (e.g., adding them, multiplying them, taking limits, etc.) results in measurable functions. The following proposition gives further information about F1 ×F2 and functions measurable with respect to it. Proposition 1. Let E ∈ F1 × F2 then for every !1 ∈ Ω 1 the set E!1 , {!2 | (!1 , !2 ) ∈ E} belongs to F2 . Consequently, for every F1 × F2 -measurable function f and every !1 the function f!1 (!2 ) , f (!1 , !2 ) 2 is F2 -measurable. Remark: E!1 and f!1 are called slicec of E and f at !1 , respectively. Proof. Fix some !1 and define a collection of sets L = {E ∈ F1 × F2 | E!1 ∈ F2 }. When E = A1 × A2 the set E!1 is either empty or equal to A2 . Thus L contains all the rectangles. On the other hand, for any sequence Ej we have [ (Ej )!1 (∪j Ej )!1 = j and (E c )!1 = (E!1 )c . Thus L is closed under countable unions and complements. Hence L is a ˙algebra, which by minimality of F1 × F2 must be equal to the latter. This shows these statement for sets. Next, a slice of a simple function f= N X ai 1Ei i=1 at !1 is itself a simple (hence measurable) function on (Ω2 , F2 ). This follows from what was just shown for slices of sets. For the general f we have f = limr→∞ fr , where fr are simple functions. Since the slice of each fr is F2 measurable and the class of F2 -measurable functions is closed under taking limits the result follows. 1.3 The product measure We now define a measure, to be denoted by P1 × P2 (or just P, for short) on the measurable space (Ω1 × Ω2 , F1 × F2 ). To capture the notion of independence, we require that P(A1 × A2 ) = P1 (A1 )P2 (A2 ), 3 ∀ A1 ∈ F1 , A2 ∈ F2 . (1) Theorem 1. There exists a unique measure P on (Ω1 × Ω2 , F1 × F2 ) that has property (1). Furthermore, for every E ∈ F1 × F2 measure P(E) satisfies Z (2) P(E) = P2 (E!1 ) P1 (d!1 ) Z = P1 (E!2 ) P2 (d!2 ). (3) Proof. Uniqueness follows from the fact that A1 × A2 is a generating psystem for F1 × F2 (see Proposition 1 in Lecture 2). We only need to show existence. We start by showing that for every E ∈ F1 × F2 the function fE (!1 ) , P2 (E!1 ) is F1 -measurable. Note that P2 (E!1 ) is well-defined by Proposition 1. Define a collection L = {E : fE is F1 -measurable}. When E = A1 × A2 the function fE (!1 ) = P2 (A2 )1A1 (!1 ), which is clearly measurable. Thus L contains all rectangles. Next, if E and F are disjoint then so are E!1 and F!1 . Consequently, fE∪F (!1 ) = fE (!1 ) + fF (!2 ) if E ∩ F = Ø . (4) This implies that L contains all finite unions of disjoint rectangles. The latter is an algebra of sets (since (A1 × A2 )c can be written as disjoint union of 3 rectangles). Finally, if Ej ր E and Ej ∈ L then f Ej ր f E (5) and therefore fE is F1 -measurable. Same argument applies to Ej ց E. All in all L is a monotone class, containing an algebra that generates F1 × F2 . So L = F1 × F2 . We now define for any E ∈ F1 × F2 Z (6) P(E) , fE (!1 )P1 (d!1 ) . It is evident that this assignment satisfies (1). Finite additivity of P follows from (4). It remains to show ˙-additivity, which in turn is equivalent to continuity. The latter follows from (5) and the MCT. 4 Thus, existence of P is established. Furthermore, definition (6) is just a restatement of (2). Regarding (3), construct another measure P′ by exchanging roles of (Ω1 , F1 , P1 ) and (Ω2 , F2 , P2 ) in (6). So constructed P′ automatically satisfies (3). Moreover, P′ also verifies (1) and hence coincides with P on a psystem of rectangles A×B. By Proposition 1 of Lecture 2 we have: P′ = P. The above discussion extends to the case of any finite number of probability spaces (Ωi , Fi , Pi ), i = 1, 2, . . . , k. In particular there exists a unique measure P on Ω = Ω1 × · · · × k such that for every collection of sets Ai ∈ Fi , P(A1 × · · · × Ak ) = P(A1 ) × · · · × P(Ak ). The corresponding ˙-algebra on Ω is the smallest ˙-algebra containing all sets of the form A1 × · · · × Ak where Ai ∈ Fi . Moreover, this extends to a countable collections of probability spaces (Ωi , Fi , Pi ), i = 1, 2, . . . , but now the measure is only defined when a finite collection of the {Ai } are not Ω k , i.e. i = 1, 2, . . . , k P(A1 × · · · × Ak × 1.4 k+1 × k+2 × · · · ) = P(A1 ) × · · · × P(Ak ). Beyond probability measures Everything in these notes extends to the case where instead of probability measures Pi , we are dealing with general measures µi , under the assumptions that the measures µi are ˙-finite. (A measure µ is called ˙-finite if the set Ω can be partitioned into a countable union of sets, each of which has finite measure.) The most relevant example of a ˙-finite measure is the Lebesgue measure on the real line. Indeed, the real line can be broken into a countable sequence of intervals (n, n + 1], each of which has finite Lebesgue measure. 1.5 The product measure on R2 The two-dimensional plane R2 is the Cartesian product of R with itself. We endow each copy of R with the Borel ˙-field B and one-dimensional Lebesgue measure. The resulting ˙-field B × B is called the Borel ˙-field on R2 . The resulting product measure on R2 is called two-dimensional Lebesgue measure, to be denoted here by 2 . The measure 2 corresponds to the natural notion of area. For example, 2 ([a, b] × [c, d]) = ([a, b]) · ([c, d]) = (b − a) · (d − c). More generally, for any “nice” set of the form encountered in calculus, e.g., sets of the form A = {(x, y) | f (x, y) ≤ c}, where f is a continuous function, 2 (A) coincides with the usual notion of the area of A. 5 Remark for those of you who know a little bit of topology – otherwise ignore it. We could define the Borel ˙-field on R2 as the ˙-field generated by the collection of open subsets of R2 . (This is the standard way of defining Borel sets in topological spaces.) It turns out that this definition results in the same ˙-field as the method of Section 1.2. 2 FUBINI’S THEOREM Fubini’s theorem is a powerful tool that provides conditions for interchanging the order of integration in a double integral. Given that sums are essentially special cases of integrals (with respect to discrete measures), it also gives conditions for interchanging the order of summations, or the order of a summation and an integration. In this respect, it subsumes results such as Corollary 1 at the end of the notes for Lecture 12. Fubini’s theorem holds under two different sets of conditions: (a) nonnegative functions g (compare with the MCT); (b) functions g whose absolute value has a finite integral (compare with the DCT). We state the two versions separately, because of some subtle differences. The two statements below are taken verbatim from the text by Adams & Guillemin, with minor changes to conform to our notation. Theorem 2. Let g : Ω1 × Ω2 → R be a nonnegative measurable function. Let P = P1 × P2 be a product measure. Then, R (a) Ω2 g(!1 , !2 ) dP2 is a measurable function of !1 . R (b) Ω1 g(!1 , !2 ) dP1 is a measurable function of !2 . (c) We have Z hZ Ω1 i g(!1 , !2 ) dP2 dP1 = Ω2 = Z ZΩ 2 hZ Ω1 i g(!1 , !2 ) dP1 dP2 g(!1 , !2 ) dP. Ω 1 × Ω2 Note that some of the integrals above may be infinite, but this is not a problem; since everything is nonnegative, expressions of the form ∞ − ∞ do not arise. Pn Proof. For simple functions g = i=1 ai 1Ei , Ei ∈ F1 × F2 statement 6 (a) follows from measurability of !1 7→ P2 (E!1 ) established in the proof of Theorem 1. For a general g consider a sequence of simple functions gr (!1 , !2 ) ր g(!1 , !2 ) ∀!1 , !2 as r → ∞. Then we have shown that Z gr (!1 , !2 ) dP2 fr (!1 ) = 2 are F1 measurable and monotonically increasing fr ր f . By the MCT Z gr (!1 , !2 ) dP2 f (!1 ) , lim r→∞ 2 Z = lim gr (!1 , !2 ) dP2 r→∞ 2 Z = g(!1 , !2 ) dP2 . (7) (8) (9) 2 Since f is a limit of measurable fr ’s – f must be measurable. By (9) the integral over 2 is also F1 measurable. This establishes (a) and (b) by symmetry. Finally (c), for a simple function g is just (2)-(3), while for a general function g we just R need to integrate (7) interchanging and lim by the MCT at will. Recall now that a function is said to be integrable if it is measurable and the integral of its absolute value is finite. 7 Theorem 3. Let g : Ω 1 × Ω2 → R be a measurable function such that Z |g(!1 , !2 )| dP < ∞, (10) Ω1 × Ω2 where P = P1 × P2 . (a) For almost all !1 ∈ Ω1 , g(!1 , !2 ) is an integrable function of !2 . (b) For almost all !2 ∈ Ω2 , g(!1 , !2 ) is an integrable function of !1 . R (c) There exists an integrable function h : Ω1 → R such that Ω 2 g(!1 , !2 ) dP2 = h(! R 1 ), a.s. (i.e., except for a set of !1 of zero P1 -measure for which Ω 2 g(!1 , !2 ) dP2 is undefined or infinite). R (d) There exists an integrable function h : Ω2 → R such that Ω1 g(!1 , !2 ) dP1 = h(! R 2 ), a.s. (i.e., except for a set of !2 of zero P2 -measure for which Ω 1 g(!1 , !2 ) dP1 is undefined or infinite). (e) We have Z hZ Ω1 Ω2 g(!1 , !2 ) dP2 dP1 = Z = Z i Ω2 hZ Ω1 i g(!1 , !2 ) dP1 dP2 g(!1 , !2 ) dP. Ω 1 × Ω2 Remarks: 1. Both Theorems remain valid when dealing with ˙-finite measures, such as the Lebesgue measure on R2 . This provides us with conditions for the familiar calculus formula Z Z Z Z g(x, y) dx dy = g(x, y) dy dx. 2. In order to apply Theorem 3, we need a practical method for checking the integrability condition (10). Here, Theorem 2 comes to the rescue. Indeed, by Theorem 2, we have Z Z Z |g(!1 , !2 )| dP2 dP1 , |g(!1 , !2 )| dP = Ω1 Ω1 × Ω2 Ω2 so all we need is to work with the right hand side, and integrate one variable at a time, possibly also using some bounds on the way. 8 Proof. By now converting from a non-negative case to integrable case should be familiar. Theorem 3 is no exception: Given a function g, decompose it into its positive and negative parts, apply Theorem 2 to each part, and in the process make sure that you do not encounter expressions of the form ∞−∞. We omit the details. 3 SOME CAUTIONARY EXAMPLES We give a few examples where Fubini’s theorem does not apply. 3.1 Nonnegativity and integrability Suppose that both of our sample spaces are the nonnegative integers: Ω 1 = Ω2 = {1, 2, . . .}. The ˙-fields F1 and F2 consist of all subsets of Ω1 and Ω2 , respectively. Then, ˙(F1 × F2 ) is composed of all subsets of {1, 2, . . .}2 . Let both P1 and P2 be the counting measure, i.e. P(A) = |A|. This means that Z Z X X f (a), h dP2 = h(b), g dP1 = A and B a∈A Z f d(P1 × P2 ) = C X b∈B f (c). c∈C Consider the function f defined by f (m, m) = 1, f (m, m + 1) = −1, and f = 0 elsewhere. It is easier to visualize f with a picture: 1 −1 0 0 ··· 0 1 −1 0 ··· 0 0 1 −1 · · · 0 0 0 1 ··· .. .. .. .. .. . . . . . So, Z Ω1 Z Ω2 f dP2 dP1 = XX n 6= 1 = f (n, m) = 0 m XX m n f (n, m) = Z Ω2 Z Ω1 f dP1 dP2 . In this example, the conditions of Fubini’s theorem fail to hold: the function f is neither nonnegative nor integrable. 9 3.2 ˙-finiteness Let Ω 1 = (0, 1), let F1 be the Borel sets, and let P1 be the Lebesgue measure. Let Ω2 = (0, 1) let F2 be the set of all subsets of (0, 1), and let P2 be the counting measure. In particular, for every infinite (countable or uncountable) subset of (0, 1), P2 (A) = ∞. Let f (x, y) = 1 if x = y, and f (x, y) = 0 otherwise. Then, Z Z Z f (x, y) dP2 (y) dP1 (x) = 1 dP1 (y) = 1, Ω1 but Z Ω2 Ω2 Z Ω1 Ω1 f (x, y) dP1 (x) dP2 (y) = Z Ω2 0 dP2 (y) = 0. In this example, the conditions of Fubini’s theorem fail to hold: the measure on (0, 1) is not ˙-finite. 10