#Contents# * table of contents {:toc} ## Definition ## A __$\sigma$-frame__ is a [[sigma-complete lattice|$\sigma$-complete lattice]] $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ with a family of dependent terms $$a:L, s:\mathbb{N} \to L \vdash a \wedge \Vee_{n:\mathbb{N}} s(n) = \Vee_{n:\mathbb{N}} a \wedge s(n)$$ representing the countably infinitary distributive property for the lattice. ## Examples ## * [[Sierpinski space]], denoted as $\Sigma$ or $1_\bot$, is the initial $\sigma$-frame. ## See also ## * [[sigma-complete lattice]] * [[distributive lattice]] * [[measure]] * [[probability measure]] ## References ## * Alex Simpson, [Measure, randomness and sublocales](https://www.sciencedirect.com/science/article/pii/S0168007211001874). * Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)