I am taking first course in Measure theory and Integration, the following question was on my mid-term exam (Though it was optional and I could attempt the other one). I have no idea as to how do I approach this problem.
Let $F$ denote the collection of all compact intervals of positive length. For each $I \in$ F, we are given a function $\Phi_{I} \in C(I\times I;[0,1])$. Let $u$ be a continuous $(-\infty, +\infty]$-valued function on $\Re$. We say $u$ is $\{\Phi_{I}\}_{I\in F}$-regular if following holds
(*) For any $I \in F$ and any $f \in C(I,\Re)$ satisfying $u|_{\partial I} \geq f|_{\partial I}$, we have
$$u(x)\geq \int_I \Phi_I(x,.)f\hspace{1.2mm}dm \hspace{10mm} \forall x\in I$$ Suppose $u$ is not identically infinity and $u$ is $\{\Phi_I\}_{I\in F}$-regular. Then show that $u^{-1}\{+\infty \}$ is nowhere dense.
Very roughly, I was thinking as follows: If we assume that $u^{-1}\{ +\infty \}$ is dense in some compact interval $I$ then by continuity of $u$ it must be infinity on whole $I$, and thereby I want to conclude that $u$ is identically infinity. I am not sure, if this approach would lead me anywhere.
I am actually not able to think how to approach this problem. Any hint/suggestion in that direction is appreciated.