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