A measure µ on a space X, defined on a sigma-algebra Σ 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 Σ, the measure of the union is the supremum of measures of elements of ${\displaystyle {\mathcal {G}}}$, i.e.:
${\displaystyle \mu \left(\bigcup {\mathcal {G}}\right)=\sup _{G\in {\mathcal {G}}}\mu (G)}$