Hausdorff measures and the Morse-Sard theorem

Let $F: U\subset \mathbb{R}^n\to\mathbb{R}^m$ be a differentiable function and $p < m$ an integer. If $k\ge1$ is an integer, $\alpha\in [0,1]$ and $F\in C^{k+(\alpha)}$, if we set $C_p(F)=\{x\in U \mid \operatorname{rank}(Df(x))\le p\}$ then the Hausdorff measure of dimension $(p+\frac{n-p}{k+\alpha})$ of $F(C_p(F))$ is zero.