Let $X$ be a metric space, $f:X \to \mathbb R$ continuous, and $g:X \to \mathbb R$ Borel measurable. If $X$ is compact, then $f$ is uniformly continuous. If $X$ is compact and $f$ locally Lipschitz-continuous, then $f$ is Lipschitz-continuous. This means compactness "strengthens" continuity. I would like to ask if this phenomenon happens for measurability, i.e.,
Does $g$ have some stronger property in case $X$ is compact?