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For $i=1,2$, let $\Sigma_i$ be two $\sigma$-algebras of subsets of some sets $S_i$. Denote by $\Sigma_1\otimes\Sigma_2$ their product $\sigma$-algebra. It is well known that any bounded, $\Sigma_1\otimes\Sigma_2$-measurable function can be approximated in the $\sup$-norm by simple functions $\sum_{k=1}^{n}a_k\cdot 1_{A_k}$, where $A_k\in \Sigma_1\otimes\Sigma_2$. Let $A\in \Sigma_1\otimes\Sigma_2$. Is it possible to approximate the characteristic function $1_A$ by simple functions of the form $1_{E\times F}$, where $E\in\Sigma_1$ and $F\in\Sigma_2$, i.e. for any $\varepsilon>0$, does there exist $n\in\mathbb{N}$, real numbers $a_1,...,a_n\in\mathbb{R}$ and sets $E_k\times F_k\in \Sigma_1\times\Sigma_2$, $k=1,...,n$ such that $$\sup_{(x,y)\in S_1\times S_2}\vert1_A(x,y)-\sum_{k=1}^{n}a_k\cdot 1_{E_k\times F_k}(x,y)\vert <\varepsilon\ ?$$

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No ... just consider $\Delta_{\mathbb R} := \{(x,x) \mid x \in \mathbb R\} \subseteq \mathbb R^2$. – martini Apr 19 '12 at 12:19
up vote 1 down vote accepted

This approximation is true if and only if $A$ is a finite union of sets of the form $A_1\times A_2$, where $A_i\in\Sigma_i$, $i\in [2]$.

Indeed, the condition is trivially sufficient, to see it's necessary take $\varepsilon=\frac 12$. We can assume that $\{E_k\}_{k=1}^n, \{F_k\}_{k=1}^n$ are pairwise disjoint families of measurable sets and $a_k\geq 0$. Then if $(x,y)\in A$, we have $$\frac 12< \sum_{k=1}^na_k\chi_{E_k}(x)\chi_{F_k}(y)< \frac 32.$$ In particular $(x,y)\in\bigcup_{k=1}^nE_k\times F_k$ so $A\subset \bigcup_{k=1}^nE_k\times F_k$. The displayed equation shows that $a_k\geq \frac 12$ for each $k\in [n]$. If $(x,y)\in E_i\times F_i$ for some $i$, we should have $|\chi_A(x,y)-a_i|<\frac 12$ and this implies $(x,y)\in A$.

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