# Measure Theory/Spaces of measurable functions

Let ${\displaystyle f:\mu \to \mathbb {R} }$ be a measurable function, where ${\displaystyle \Omega }$ is a topological space and ${\displaystyle \mu }$ is a measure on the Lebesgue ${\displaystyle \sigma }$-algebra of ${\displaystyle \Omega }$. Then the essential support of ${\displaystyle f}$ is the set
${\displaystyle \operatorname {esssupp} f:={\overline {\{x\in \Omega |\forall \epsilon >0:\mu (B_{\epsilon }(x)\cap \{y\in \Omega |f(y)\neq 0\})>0\}}}}$.
The essential support ${\displaystyle \operatorname {esssupp} }$ is not to be confused with the essential supremum ${\displaystyle \operatorname {esssup} }$.