In the book Measure Theory and Fine Properties of Functions of Evans and Gariepy has the theorem:
Theorem (Area Formula): Let $f: \mathbb{R}^n \longrightarrow \mathbb{R}^m$ be Lipschitz, $n\leq m$. Then for each $\mathcal{L}^n$-measurable subset $A\subset \mathbb{R}^n$, $\int_A Jfdx=\int_{\mathbb{R}^m} \mathcal{H}^0(A\cap f^{-1}(y))d\mathcal{H}^n(y)$.
Where, $\mathcal{H}^n$ and $\mathcal{H}^m$ are Hausdorff measure. $\mathcal{H}^0$ is counting measure. $J(f)$ is the Jacobian of $f$.
After the theorem has the remark,
Remark: Using the Area Formula, we see $f^{-1}(y)$ is at most countable for $\mathcal{H}^n$ a.e $y\in \mathbb{R}^m$.
My question: How to prove this remark?
Fix a compact set $K$ in ${\bf{R}}^{n}$, then $\displaystyle\int_{K}Jf(x)dx<\infty$ and hence $\displaystyle\int_{K}\mathcal{H}^{0}(K\cap f^{-1}(y))d\mathcal{H}^{n}(y)<\infty$, so for $\mathcal{H}^{n}$-a.e. $y$ in $K$, $\mathcal{H}^{0}(K\cap f^{-1}(y))<\infty$, so $K\cap f^{-1}(y)$ has finitely many elements. Now write ${\bf{R}}^{n}$ as the union of an increasing sequence of compact sets, and note that the countable union of null sets is still null, and the countable union of finite sets is of at most countable, the assertion follows.