Are there Lebesgue-measurable functions not almost everywhere equal to a continuous function

Question

This is why I originally meant to ask with Are there Lebesgue-measurable functions non-continuous almost everywhere?

Does there exist a function $f\colon [0,1]\to\mathbb{R}$ such that:

$f$ is Lebesgue measurable; and
For every continuous $g\colon [0,1]\to\mathbb{R}$, the set of points where $f(x)\neq g(x)$ has positive measure?

The question listed as duplicate is actually a stronger statement. For this one, $f = 1_{[0,1/2]}$ is a counterexample. — Nate Eldredge, Dec 10 '11 at 02:07

score 5 · Answer 1 · edited Apr 13 '17 at 12:19

5

Yes. Fix any measurable set $A$ such that both $A$ and its complement have non-null intersection with each nonempty open interval. Examples are discussed here. Then the characteristic function of $A$ is as desired, since removing a null set does not change this intersection property, which rules out having a continuous extension.

edited Apr 13 '17 at 12:19

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answered Sep 23 '11 at 20:30

user83827

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Are there Lebesgue-measurable functions not almost everywhere equal to a continuous function

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