Measure of boundary in $\mathbb R^n$

Question

I saw that the measure of the boundary of a regular open set in $\mathbb{R}^n$ is zero, so how can we talk about the integral on this boundary (for me it must be equal to zero always )? I saw that in Sobolev spaces, Green formula, etc.)

Answer 1

The integral on the boundary is done with respect to the $n-1$-dimensional Lebesgue measure, as the boundary of a regular set in $\mathbb R^n$ is an $n-1$-dimensional object.

For instance, in $\mathbb R^2$, the boundary of a disk is a circle, which has zero area (i.e. $2$-dimensional Lebesgue measure), but positive length (i.e. $1$-dimensional Lebesgue measure).