Let $(X,d)$ be a complete metric space, $\mu$ a finite Borel measure of $(X,d)$ and $f \colon X \to \mathbb{R}$ measurable.
Lusin's theorem gives us that for each $\varepsilon>0$ there exists a closed set $C$ such that $\mu(X\backslash C)<\varepsilon$ and $f \mid_C$ is continuous.
For each $\varepsilon >0$, can we find a measurable set $C$ such that $\mu(X\backslash C)<\varepsilon$ and $f \mid_C$ is uniformly continuous?
I know that the result is true if we also assume that $(X,d)$ is separable: in this case, we can assume that Lusin's $C$ is compact and then we can rely on Heine-Cantor theorem. But what about the general case?