A probability distribution is a measure used in probability theory whose integral over some subspace of a measurable space is regarded as assigning a probability for some event to take values in this subset.

it is positive: $\forall U \subset X : \int_U d\rho \geq 0$;

it is normalized: $\int_X d\rho = 1$.

On $\sigma$-frames

In measure theory, a probability measure or probability distribution on a $\sigma$-frame or more generally a $\sigma$-completedistributive lattice$(L, \leq, \bot, \vee, \top, \wedge, \Vee)$ is a probability valuation$\mu:L \to [0, 1]$ such that the elements are mutually disjoint and the probability valuation is denumerably/countably additive

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