---
tag: QDDRDW9PXS
kind: concept
name: Standard probability space
msc: [28A05, 60A10, 28D05]
specializes: [0NFPJVJJHX]
sources:
  - url: https://en.wikipedia.org/wiki/Standard_probability_space
same_as:
  wikidata: Q7598372
  wikipedia: Standard probability space
---

## Summary

A standard probability space is a [[probability space|0NFPJVJJHX]] that looks like a simple model after null sets are removed.
The model is an interval with Lebesgue measure, a countable set of atoms, or a combination of both.
The theory goes back to von Neumann in 1932 and was shaped by Rokhlin, who called these spaces Lebesgue spaces.
Most probability spaces of analysis are standard, and on standard spaces conditional probabilities and disintegrations behave well.
For this reason ergodic theory and probability often assume their spaces to be standard.

## Explanations

### L1 human

A standard probability space is one that, after ignoring events of probability zero, looks like a simple model.
The model is a piece of the unit interval where probability is length, plus some single points that carry positive probability.
Picking a random point of a square, or flipping infinitely many coins, gives a standard space.
Strange spaces built with the axiom of choice can fail to be standard, and they are the exception.