Regular measure

From Wikipedia, the free encyclopedia

In mathematics, a regular measure on a topological space is one for which every measurable set is "approximately open" and "approximately closed".

1 Definition
2 Example
3 Reference
4 See also

[edit] Definition

Let $(X, \mathcal{T})$ be a topological space, and let $\mathcal{A}$ be a sigma algebra on $X$ that contains the topology $\mathcal{T}$ (so every open set (and hence every closed set) is measurable). A measure $μ$ on $(X, \mathcal{A})$ is called regular if, for every measurable set $A \in \mathcal{A}$ and $\varepsilon > 0$ , there exists a closed set $C$ and an open set $U$ such that $C \subseteq A \subseteq U$ and $\mu (U \setminus C) < \varepsilon$ .