τ-additivity

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

A measure or set function $\mu$ on a space $X$ whose domain is a sigma-algebra $\Sigma$ 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 $\Sigma ,$ the measure of the union is the supremum of measures of elements of ${\mathcal {G}};$ that is,: $$\mu \left(\bigcup {\mathcal {G}}\right)=\sup _{G\in {\mathcal {G}}}\mu (G).$$

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.