I am looking for (the name of) a class of functions from $\mathbf{R}^2$ $(\mathbf{R}^n)$ to {0, 1} that are integrable.

Let $f$ be in this class and $E$ be the set of all points where $f$ is equal to $1$. Then for all $x\in E$, the following holds for all $r>0$: $$ \operatorname{area}( B(x, r) \cap E ) > 0. $$ The same holds for the points where $f$ is $0$.

That is, $f$ can not arbitrarily be changed on a null set and still remain in this class. The goal is to be able to talk about point-wise values of solutions to optimization problems involving integrals.

Does this class (or a related one) have a name?