On a Hausdorff topological space $X$ with a sigma algebra $ Σ$ at least as fine as the Borel sigma algebra,
- a measure $\mu$ is said to be inner regular, if for every set $A \in Σ$, $\mu(A) = \sup \{ \mu(K) | \text{ compact }K \subseteq A \}$.
- a measure is said to be tight, if for all $ε > 0$, there is some compact subset $K$ of $X$ such that $μ(X - K) < ε$.
Wikipedia says that a measure is inner regular iff it is tight. I was wondering why? Is it still true when the topological space $X$ is not necessarily Hausdorff? References are also appreciated!
Thanks and regards!