---
tag: JC0SZT6453
kind: definition
name: Young measure
msc: [49J45, 28A33]
defines: 6TQSVJQDEH
uses: [059TJN0NZ1, YYNCJSR0PJ]
sources:
  - url: https://en.wikipedia.org/wiki/Young_measure
---

## Definition

Let $n, m \ge 1$, and let $\Omega \subseteq \R^n$ be a Lebesgue measurable set.
Let $C_0(\R^m)$ be the space of continuous functions $\varphi \colon \R^m \to \R$ that tend to $0$ at infinity.
A *Young measure* on $\Omega$ with values in $\R^m$ is a family $(\nu_x)_{x \in \Omega}$ of Borel [[probability measures|059TJN0NZ1]] on $\R^m$.
It must satisfy one condition: for every $\varphi \in C_0(\R^m)$ the function
$$x \mapsto \int_{\R^m} \varphi \, d\nu_x$$
is Lebesgue [[measurable|YYNCJSR0PJ]] on $\Omega$.

## Remarks

A sequence of measurable $u_k \colon \Omega \to \R^m$ *generates* $(\nu_x)$ if $\varphi(u_k) \to \int \varphi \, d\nu_x$ weakly-$*$ in $L^\infty(\Omega)$ for every $\varphi \in C_0(\R^m)$.
The fundamental theorem on Young measures says that every sequence bounded in $L^\infty(\Omega; \R^m)$ has a subsequence that generates a Young measure.
Example: a measurable $u \colon \Omega \to \R^m$ gives the Young measure $\nu_x = \delta_{u(x)}$, since $\int \varphi \, d\nu_x = \varphi(u(x))$ is measurable.
Example: $u_k(x) = \sgn(\sin(2 \pi k x))$ on $(0,1)$ generates the constant family $\nu_x = \frac12 (\delta_{-1} + \delta_1)$.
Non-example: if $g \colon \Omega \to \R^m$ is not measurable, then $\nu_x = \delta_{g(x)}$ is not a Young measure.
Some open $U$ has a non-measurable preimage, and $1_U$ is an increasing limit of functions $\varphi_j \in C_0(\R^m)$, so some $\varphi_j \circ g$ is not measurable.