Approximation of jointly measurable, bounded function by sum of tensor products

Question

The following result is claimed by Oksendal in his proof of the Markov property for Ito diffusions (see the proof of Theorem 7.1.2). Let $(X_1, \mathcal{A}_1, \mu_1)$ and $(X_2, \mathcal{A}_2, \mu_2)$ be two $\sigma$-finite measure spaces. Let $f: X_1 \times X_2 \to \mathbb{R}$ be a bounded, Borel measurable function with respect to the product $\sigma$-algebra $\mathcal{A}_1 \times \mathcal{A}_2$. That is, $f^{-1}(B) \in \mathcal{A}_1 \times \mathcal{A}_2$ for every Borel measurable set $B \subseteq \mathbb{R}$. Then, we can approximate $f$ pointwise (everywhere) by functions of the form $$\sum_{k=1}^m \psi_k(x_1) \phi_k(x_2),$$ where each $\psi_k:X_1 \to \mathbb{R}$ is bounded and Borel measurable, and similarly for each $\phi_k$.

Any ideas of how to prove this and whether or not it is even true?

Answer 1

Broad outline for almost everywhere convergence:

First assume that the measures are finite. Then $f \in L^{2}(X_1 \times X_2)$. Prove that finite sums of the type mentioned in the question form dense subset of $L^{2}(X_1 \times X_2)$ by showing that if a function in $L^{2}(X_1 \times X_2)$ ir orthogonal to all functions of the form $\psi(x_1)\phi(x_2)$ then it is $0$ a.e. [This is an easy application of Fubini's Theorem]. Convergence in $L^{2}$ implies almost everywhere convergence of a subsequence. This proves the result in this case. For the general case there exist disjoint rectangles $R_i$ such that $(\mu_1 \times \mu_2) (R_i) <\infty$ for all $i$. Apply the result to the restriction of the measure spaces to $R_i$.

Answer 2

Lets consider the semi-algebra $\mathcal{A}:= \{ A_1\times A_2 | \ A_1\in \mathcal{A}_1 \ \& \ A_2\in \mathcal{A}_2 \}$ that generates the $\sigma$-algebra $\mathcal{A}_1 \otimes \mathcal{A}_2$. Define $\mathcal{B}:= \{ B_1 \dot\cup B_2 \cdots \dot\cup B_n | \ B_1,B_2, \ldots, B_n\in \mathcal{A}, \ n\in \mathbb{N}\}$, i.e. the set of all finite, disjoint unions of elements of $\mathcal{A}$. Then $\mathcal{B}$ is an algebra generating the $\sigma$-algebra $\mathcal{A}_1 \otimes \mathcal{A}_2$. Then if $A\in \mathcal{A}_1 \otimes \mathcal{A}_2$, then $\forall \ \epsilon >0$, $\exists \ B \in \mathcal{B}$ such that $\mu_1\otimes \mu_2 (A \bigtriangleup B)<\epsilon$. In other words characteristic functions of the for $\chi_{A\times B}(a,b) = \chi_A(a) \cdot \chi_B(b)$ approaches in $L^1(\mu_1\otimes \mu_2)$ characteristic functions of the general form. More specifically, you can find a sequence of such characteristic functions that converges to the general characteristic function in $L^1(\mu_1\otimes \mu_2)$. Thus there exist a subsequence s.t. it converges pointwise a.e. to the general characteristic function. Now you are basically done, since you can now prove it for simple functions, then positive measurable functions and finally for general functions.