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