Let $\Omega$ be an open (non empty) set and $u:\Omega \subset \mathbb{R}^N \to \mathbb{R}^M$ be a function such that the Hausdorff dimension of its graph is $N$.

Let $\tilde u = u$ a.e. Is it true that the Hausdorff dimension of the graph of $\tilde u$ is also $N$?