Whenever editing is allowed on the nLab again, this article should be ported over there.

Definition

In set theory

In measure theory, a measure on a $\sigma$-frame or more generally a $\sigma$-completedistributive lattice$(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a valuation$\mu:L \to [0, \infty]$ such that the elements are mutually disjoint and the probability valuation is denumerably/countably additive

$\forall s\in L^\mathbb{N}. \forall m \in \mathbb{N}. \forall n \in \mathbb{N}. (m \neq n) \wedge (s(m) \wedge s(n) = \bot)$