Measure of Cross Section

• categories: Measure Theory, Definition

Measure of Cross Section

Definition: Given a measurable set $A\subseteq {\mathbb{R}}^{n+m}$ and a fixed point $x\in {\mathbb{R}}^n$, the cross section of $A$ at $x$ is defined as:

$$A_x=\{y\in {\mathbb{R}}^m:(x,y)\in A\}$$

The measure of the cross section is then the Lebesgue measure of $A_x$ in ${\mathbb{R}}^m$, denoted by:

$${\mu }_m(A_x)={\int }_{{\mathbb{R}}^m}{\chi }_{A_x}(y)dy$$

Intuition: The measure of cross sections provides a way to slice a higher-dimensional set into lower-dimensional pieces

Key Application: Fubini’s Theorem and Tonelli’s Theorem