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Density of Tensor Products

Mathematics Asked by Jacob Denson on September 8, 2020

Let $X$ and $Y$ be measure spaces. Under what conditions is it true that the space of all finite simple functions of the form

$$ sum_{i=1}^N a_i mathbf{I}_{E_i times F_i} $$

forms a dense subspace of the space of all elements of the mixed norm space $L^p(Y)L^{infty}(X)$. In other words, under what conditions is true that for any function $f$ with

$$ sup_{x in X} int_Y | f(x,y) |^p dy < infty $$

we can find a simple function for each $varepsilon > 0$ such that

$$ sup_{x in X} int_Y | f(x,y) – sum a_i mathbf{I}_{E_i times F_i}(x,y) |^p dy < varepsilon. $$

where the suprema are of course treated as almost everywhere suprema ala the $L^infty$ norm.

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