---
tag: Y98C5T956Q
kind: concept
name: Measure with density
msc: [28A25, 28A10]
specializes: [PR11BZQQPG]
sources:
  - url: https://en.wikipedia.org/wiki/Radon–Nikodym_theorem
---

## Summary

Given a [[measure space|P15VQS5DJ9]] $(X, \Sigma, \mu)$ and a measurable function $f \ge 0$, the measure with density $f$ gives each set $A$ the mass $\int_A f \, d\mu$.
It is written $f \mu$ or $f \, d\mu$.
It is the basic way to build new measures from old ones.
The Radon–Nikodym theorem says that, for σ-finite measures, every measure absolutely continuous with respect to $\mu$ is of this form.

## Explanations

### L1 human

Start with length on the real line and a weight function $f$.
The new measure of an interval is the area under the graph of $f$ over that interval.
Where $f$ is large, the new measure packs more mass.
With $f(x) = e^{-x^2/2} / \sqrt{2\pi}$ and Lebesgue measure, one gets the standard normal distribution.