In mathematics, in the field of measure theory, τ-additivity is a certain property of measures on topological spaces.

A measure or set function ${\displaystyle \mu }$ on a space ${\displaystyle X}$ whose domain is a sigma-algebra ${\displaystyle \Sigma }$ is said to be τ-additive if for any upward-directed family ${\displaystyle {\mathcal {G}}\subseteq \Sigma }$ of nonempty open sets such that its union is in ${\displaystyle \Sigma ,}$ the measure of the union is the supremum of measures of elements of ${\displaystyle {\mathcal {G}};}$ that is,:

$${\displaystyle \mu \left(\bigcup {\mathcal {G}}\right)=\sup _{G\in {\mathcal {G}}}\mu (G).}$$

See also

• Net (mathematics) – A generalization of a sequence of points
• Sigma additivity – Mapping functionPages displaying short descriptions of redirect targets
• Valuation (measure theory) – map in measure or domain theoryPages displaying wikidata descriptions as a fallback

References

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