τ-additivity
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In mathematics, in the field of measure theory, τ-additivity is a certain property of measures on topological spaces.
A measure or set function [math]\displaystyle{ \mu }[/math] on a space [math]\displaystyle{ X }[/math] whose domain is a sigma-algebra [math]\displaystyle{ \Sigma }[/math] is said to be τ-additive if for any upward-directed family [math]\displaystyle{ \mathcal{G} \subseteq \Sigma }[/math] of nonempty open sets such that its union is in [math]\displaystyle{ \Sigma, }[/math] the measure of the union is the supremum of measures of elements of [math]\displaystyle{ \mathcal{G}; }[/math] that is,: [math]\displaystyle{ \mu\left(\bigcup \mathcal{G}\right) = \sup_{G\in\mathcal{G}} \mu(G). }[/math]
See also
- Net (mathematics) – A generalization of a sequence of points
- Sigma additivity
- Valuation (measure theory)
References
- Fremlin, D.H. (2003), Measure Theory, Volume 4, Torres Fremlin, ISBN 0-9538129-4-4.
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