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 ${\displaystyle \mu }$ 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,:

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