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 : X× 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

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.