#Contents# * table of contents {:toc} Whenever editing is allowed on the [[nLab:HomePage|nLab]] again, this article should be ported over there. ## Definition ## ### In set theory ### In measure theory, a __$\sigma$-continuous valuation__ on a [[sigma-complete lattice|$\sigma$-complete lattice]] $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a [[valuation (measure theory)|valuation]] $\mu:L \to [0, \infty]$ such that the $\sigma$-continuity condition is satisfied $$\forall s \in L^\mathbb{N}. (\forall n \in\mathbb{N}. s(n) \leq s(n + 1)) \implies \left(\mu(\Vee_{n:\mathbb{N}} s(n)) \leq \sup_{n:\mathbb{N}} \mu(s(n)) \right)$$ ### In homotopy type theory ### In measure theory, a __$\sigma$-continuous valuation__ on a [[sigma-complete lattice|$\sigma$-complete lattice]] $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a [[valuation (measure theory)|valuation]] $\mu:L \to [0, \infty]$ 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 ## * [[valuation (measure theory)]] * [[sigma-continuous probability valuation]] * [[measure]] ## References ## * Alex Simpson, [Measure, randomness and sublocales](https://www.sciencedirect.com/science/article/pii/S0168007211001874).