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Let $(X, \mathscr{S}, \mu)$ and $(Y, \mathscr{T}, \lambda)$ be two $\sigma$-finite measure spaces, give $X\times Y$ the product measure, then is it true that for any positive measurable function $f:X\times Y\rightarrow [0, \infty]$, there is a sequence of simple functions $g_{k}$, each is of the form $\sum_{i,j=1}^{s}a_{ij}\chi_{A_{i}}\chi_{B_{j}}$ (finite sum, where $A_{i}$ and $B_{j}$ are measurable sets in $X$ and $Y$ respectively), that converges poinwisely to $f$ (or at least a.e. to $f$)?

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