# nLab probability distribution

### Context

#### Measure and probability theory

measure theory

probability theory

## Measure theory

• von Neumann algebra

• geometric measure theory

• ## Probability theory

• probability space

• probability distribution

• state

• ## Information geometry

• information geometry

• information metric

• Wasserstein metric

• ## Thermodynamics

• thermodynamics

• ergodic theory

• ## Theorems

• Riesz representation theorem
• # Contents

## Idea

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.

## Definition

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: $\int_X d\rho = 1$.

## Properties

The collection of all probability distributions on a measurable space carries various metric structures that are studied in information geometry:

## Examples

Last revised on May 1, 2016 at 21:06:25. See the history of this page for a list of all contributions to it.