Let be $u$ a numerical function defined over $\Omega$, with $u$ measurable, and let be $(O_i)_{i\in I}$ a family of all open sub-sets $O_i$ of $\Omega$, such that $u=0$ almost always except on a set of measure $0$ in $O_i$. Let be $O = \cup_{i\in I}O_i$ (Then $u=0$ almost always except on a set of measure in $O$).
I need show that: If $u$ is continuos then supp $u = \Omega\setminus O = closure(\{x \in \Omega; u(x) \neq 0\})$.
How I will be able to making this?