1.7. Outer measures, pre-measures, product measures 161 1.7.4. Product measure. Given two sets X and Y , one can form their Cartesian product X × Y = {(x, y) : x X, y Y }. This set is naturally equipped with the coordinate projection maps πX : X×Y X and πY : Y Y defined by setting πX (x, y) := x and πY (x, y) := y. One can certainly take Cartesian products X1 × . . . × Xd of more than two sets, or even take an infinite product α∈A Xα, but for simplicity we will only discuss the theory for products of two sets for now. Now suppose that (X, BX ) and (Y, BY ) are measurable spaces. Then we can still form the Cartesian product X × Y and the projection maps πX : X × Y X and πY : X × Y Y . But now we can also form the pullback σ-algebras πX (BX ) := {πX1(E) : E BX } = {E × Y : E BX } and πY (BY ) := {πY −1(E) : E BY } = {X × F : F BY }. We then define the product σ-algebra BX ×BY to be the σ-algebra generated by the union of these two σ-algebras: BX × BY := πX(BX ) πY (BY ) . This definition has several equivalent formulations: Exercise 1.7.18. Let (X, BX ) and (Y, BY ) be measurable spaces. (i) Show that BX ×BY is the σ-algebra generated by the sets E×F with E BX , Y BY . In other words, BX ×BY is the coarsest σ-algebra on X × Y with the property that the product of a BX -measurable set and a BY -measurable set is always BX × BY measurable. (ii) Show that BX ×BY is the coarsest σ-algebra on X × Y that makes the projection maps πX,πY both measurable morphisms (see Re- mark 1.4.33). (iii) If E BX × BY , show that the sets Ex := {y Y : (x, y) E} lie in BY for every x X, and similarly that the sets Ey := {x X : (x, y) E} lie in BX for every y Y . (iv) If f : X × Y [0, +∞] is measurable (with respect to BX × BY ), show that the function fx : y f(x, y) is BY -measurable for every x X, and similarly that the function f y : x f(x, y) is BX - measurable for every y Y . (v) If E BX × BY , show that the slices Ex := {y Y : (x, y) E} lie in a countably generated σ-algebra. In other words, show that there exists an at most countable collection A = AE of sets (which can depend on E) such that {Ex : x X} A. Conclude, in particular, that the number of distinct slices Ex is at most c, the
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