I have this question: suppose that $f$ is a function in $L^1 \cap L^\infty$ (the Lebesgue measure is used tacitly) such that $f \geq 0$ a.e. And $f$ is not the zero function a.e. Does there exist a rectangle such that the essential infimum of $f$ on such a rectangle is strictly positive?
I feel like the answer is negative, since there exist sets of positive measure which do not contain any rectangle. But I am not sure if I am missing something in this reasoning.