Sigma-additive set function: Difference between revisions

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Suppose that in addition to a sigma algebra <math display=inline>\mathcal{A},</math> we have a [[Topological space|topology]] <math>\tau.</math> If for every [[Directed set|directed]] family of measurable [[open set]]s <math display=inline>\mathcal{G} \subseteq \mathcal{A} \cap \tau,</math>

<math display=block>\mu\left(\bigcup \mathcal{G} \right) = \sup_{G\in\mathcal{G}} \mu(G),</math>

we say that <math>\mu</math> is <math>\tau</math>-additive. In particular, if <math>\mu</math> is [[Inner regular measure|inner regular]] (with respect to compact sets) then it is &<math>\tau;</math>-additive.<ref name=Fremlin>D. H. Fremlin ''Measure Theory, Volume 4'', Torres Fremlin, 2003.</ref>

==Properties==