Let $X$ be a Hausdorff space, $\mathcal{B}(X)$ the Borel $\sigma$-algebra and $\mu : \mathcal{B}(X) \to [0, \infty]$ a measure.
Consider the following properties:
(1) $\forall A \in \mathcal{B}(X): \mu(A) = \sup \left\{ \mu(K) \mid K \subseteq A, K \text{ compact} \right\}$
(2) $\forall \varepsilon > 0 \, \exists K \subseteq X$ compact: $\mu\left(X \setminus K\right) < \varepsilon$
If $\mu$ satisfies (1) then some authors say that $\mu$ is "tight" or "inner regular" or "inner regular w.r.t. the compact sets". Often those authors that reserve the notion "inner regular" for "inner regular w.r.t. the closed sets" call $\mu$ "tight" if it satisfies (1). Other authors, mostly probabilists, refer to a finite measure (probability measure) $\mu$ as "tight" if it satisfies (2). (2) seems not to be an interesting property for measures $\mu$ that are not totally finite (but at least finite on compact sets). As an example, the Lebesgue measure on $\mathbb{R}$ is "tight" in the sense of (1) but not "tight" in the sense of (2).
If $\mu$ is finite then (1) $\Rightarrow$ (2). It is also folklore that if $X$ is metrizable and $\mu$ finite then (1) $\Leftrightarrow$ (2) (see also here).
Questions:
- Can the assumption on $X$ being metrizable be weakened, e.g. perfectly normal or even weaker?
- Are there interesting applications in which tightness in the sense of property (1) is of interest when $\mu$ is finite, but (2) is not applicable?