PomeDefinition 2.2.1.901C72102EF9 A Radon measure on $\Omega $ is a Borel measure $\mu $ on $\Omega $ which is finite on compact sets and satisfies, for every Borel set $E\subset \Omega $,
\[ \mu (E)=\inf _{E\subset U,\ U\text{ open}}\mu (U) =\sup _{K\subset E,\ K\text{ compact}}\mu (K). \]
The two equalities are called outer and inner regularity of $\mu $. A measurable function is locally integrable with respect to $\mu $ if its absolute value has finite integral over every compact subset of $\Omega $.
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