Consider a function $f(x,y): [0,1]^2 \to [0,\infty)$ continuous almost everywhere, for which there is no $A \subset [0,1]$ such that $0<\mu(A)<1$ and $\int_{A \times A^c} f(x,y)dxdy=0$.

Is it true that for every $\varepsilon >0$, $\inf_{A : \varepsilon<\mu (A)<1-\varepsilon} \int_{A \times A^c} f(x,y)dxdy>0$?

  • $\begingroup$ It looks like this problem would be susceptible to textbook functional analysis tools (I'm thinking Banach–Alaoglu). You would probably get a quick answer on math.stackexchange.com. $\endgroup$ – Dap Feb 15 at 17:00

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