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

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

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