By Lusin-theorem, every measurable function is continuous on a compact subset of their domain, which is endowed with sub-space topology and its complement is nearly a Nullset.
Now suppose we have the indicator function of a fat cantor set. The fat cantor set has measure one half and is closed. The indicator function is constant 1 and will be continuous on the fat cantor set. However, it’s complement has a fixed measure of one half, which cannot be made arbitrary close to a null set. How can I find a close set that satisfies the Lusin theorem?