If two functions are equal almost everywhere on a measurable set, then they are equal on it?

Question

Suppose $f$ and $g$ are continuous functions on a measurable set $E$. Is it true if $f=g$ almost everywhere on $E$, then $f=g$ on E? Here, measurable means "Lebesgue measurable."

When it is a closed interval, it works. But after it I can't go further.

How can I prove or disprove this?

Answer 1

No; consider the set $E = [0,1] \cup \{2\}$. Two functions which agree on $[0,1]$ but differ on the single point $2$ are almost everywhere equal, but not equal.