# Contents

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 $\sigma$-complete distributive 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

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

### In homotopy type theory

In measure theory, a measure on a $\sigma$-frame or more generally a $\sigma$-complete distributive lattice $(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a valuation $\mu:L \to [0, \infty]$ with

• a family of dependent terms
$s:\mathbb{N} \to L \vdash \alpha(s):\left(\prod_{m:\mathbb{N}} \prod_{n:\mathbb{N}} (m \neq n) \times (s(m) \wedge s(n) = \bot)\right) \times \left(\mu(\Vee_{n:\mathbb{N}} s(n)) = \sum_{n:\mathbb{N}} \mu(s(n)) \right)$

representing the mutually disjoint elements condition and the denumerably/countably additive condition.