# 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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