Integral operators

1 Product measures Let (X,A , μ) be a σ-finite measure space. Then with A ⊗ A the product σalgebra and μ ⊗ μ the product measure on A ⊗A , (X ×X,A ⊗A , μ⊗ μ) is itself a σ-finite measure space. Write Fx(y) = F (x, y) and F (x) = F (x, y). For any measurable space (X ′,A ′), it is a fact that if F : X×X → X ′ is measurable then Fx is measurable for each x ∈ X and F y is measurable for each y ∈ X. Suppose that F ∈ L (X × X), F : X → C. Fubini’s theorem tells us the following. 3 There are sets N1, N2 ∈ A with μ(N1) = 0 and μ(N2) = 0 such that if x ∈ N c 1 then Fx ∈ L (X) and if y ∈ N c 2 then F y ∈ L (X). Define