nLab sigma-continuous probability valuation

Definition

In measure theory, a $\sigma$-continuous probability valuation on a $\sigma$-complete lattice $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a probability valuation $\mu:L \to [0, 1]$ such that the $\sigma$-continuity condition is satisfied: for all sequences $s:\mathbb{N} \to L$, if $s(n) \leq s(n + 1)$ for all natural numbers $n \in\mathbb{N}$, then

$$\mu(\Vee_{n:\mathbb{N}} s(n)) \leq \sup_{n:\mathbb{N}} \mu(s(n))$$

See also

• unit interval

• sigma-continuous valuation

• probability valuation

• probability measure

References

• Alex Simpson, Measure, randomness and sublocales, Annals of Pure and Applied Logic, Volume 163, Issue 11, November 2012, Pages 1642-1659. (doi:10.1016/j.apal.2011.12.014)