Let be $u$ a numerical function defined over $\Omega$, with $u$ measurable, and let $(O_i)_{i\in I}$ be 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 $O = \cup_{i\in I}O_i$ (Then $u=0$ almost always except on a set of measure in $O$). How can I define in explicit form the set $\Omega\setminus O$?
I trying this ...
$\Omega\setminus O = \{x\in \Omega; (u(x)=0 \text{ and } x\in K\subset \Omega, \text{ where } K \text{ is closed}) \text{ or } (u(x)\neq 0)\}$.
This is correct?