# Uniform lower bound on integral over sets of the form $A \times A^c$

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

• 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. – Dap Feb 15 at 17:00