Let $u$ be a upper semicontinuous function on a compact set $K$ in $\mathbb R^d$. Define a space of continuous function dominating $u$ by $$A = \{\phi \in C(K): \phi \ge u\}.$$ [Q.] Is the following inequality true? $$\inf_{\phi \in A} \int_K (\phi - u) dx = 0?$$

It seems true, if $d = 1$. Indeed, one can simply take linear interpolation from each discontinuous point of $u$ to its small neighborhood. However, I do not have idea when $d\ge 2$. Does it related to the geometry of discontinuous set of $u$?