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 µ on a space X, defined on a sigma-algebra Σ is said to be τ-additive, if for any upward-directed family \mathcal{G}\subseteq \Sigma of nonempty open sets, such that its union is in Σ, the measure of the union is the supremum of measures of elements of \mathcal G, i.e.:

\mu\left(\bigcup \mathcal{G}\right)=\sup_{G\in\mathcal{G}}\mu(G)

References[edit]

  • Fremlin, D.H. (2003), Measure Theory, Volume 4, Torres Fremlin, ISBN 0-9538129-4-4 .