title: Τ-additivity
text: # 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,:
  
  $\mu \left(\bigcup {\mathcal {G}}\right)=\sup _{G\in {\mathcal {G}}}\mu (G).$
  
  