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.
Often a probability density.
A probability distribution is a measure $\rho$ on a measurable space $X$ such that
it is positive: $\forall U \subset X : \int_U d\rho \geq 0$;
it is normalized: $\int_X d\rho = 1$.
The collection of all probability distributions on a measurable space carries various metric structures that are studied in information geometry: