Infinite dimensional integral inequality

real-analysis functional-analysis

So, first thing to try is to set

$$f(x,t) = \sum_{j = 1}^N g_j(x) 1_{F_j}(y)$$

for pairwise disjoint $F_j$. For this the inequality is easy to verify.

Now we would like to take limits, but the question is: Are the measurable functions pointwise limits of function of the form of $f$? It turns out this is the case as has been shown by Nate Eldredge on a question of mine (I tried to prove the same). Sequence of measurable functions

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