τ-additivity

In mathematics, in the field of measure theory, τ-additivity is a certain property of measures on topological spaces.

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.:

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

See also

• Sigma additivity
• Valuation (measure theory)
• Net (mathematics)

References

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