TITLE: Inner regular measures and tight measures
QUESTION [6 upvotes]: 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!

REPLY [4 votes]: Wikipedia is wrong because its definition of a tight measure as "for all ε>0, there is some compact subset K of X such that μ(X−K)<ε" fails to note that the measure must be a probability (or totally finite) measure. (Parthasarathy, 1967, Probability Measures on Metric Spaces, p. 26, fn 1.)  Otherwise, a measure μ is said to be tight, if for every set A∈Σ, μ(A)=sup{μ(K)| compact K⊆A} and inner regular, if for every set A∈Σ, μ(A)=sup{μ(F)| F closed  and F⊆A}.