Measure Theory/Spaces of measurable functions

Definition (essential support):

Let $f:\mu \to \mathbb {R}$ be a measurable function, where $\Omega$ is a topological space and $\mu$ is a measure on the Lebesgue $\sigma$-algebra of $\Omega$. Then the essential support of $f$ is the set

$\operatorname {esssupp} f:={\overline {\{x\in \Omega |\forall \epsilon >0:\mu (B_{\epsilon }(x)\cap \{y\in \Omega |f(y)\neq 0\})>0\}}}$.

The essential support $\operatorname {esssupp}$ is not to be confused with the essential supremum $\operatorname {esssup}$.