#Contents# * table of contents {:toc} ## Definition ## In measure theory, a __$\sigma$-continuous probability valuation__ on a [[sigma-complete lattice|$\sigma$-complete lattice]] $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a [[probability valuation]] $\mu:L \to [0, 1]$ with * a family of dependent terms $$s:\mathbb{N} \to L \vdash c(s):\left(\prod_{n:\mathbb{N}} s(n) \leq s(n + 1)\right) \to \left(\mu(\Vee_{n:\mathbb{N}} s(n)) \leq \sup_{n:\mathbb{N}} \mu(s(n)) \right)$$ representing the $\sigma$-continuity condition. ## See also ## * [[unit interval]] * [[sigma-continuous valuation]] * [[probability valuation]] * [[probability measure]] ## References ## * Alex Simpson, [Measure, randomness and sublocales](https://www.sciencedirect.com/science/article/pii/S0168007211001874).