Is this result in general true for lebesgue measurable sets

Question

I know that if $f,g$ and continuous on $[a,b]$ and if $f=g$ a.e. on $[a,b]$, then in fact $f=g$ on $[a,b]$. Moreover, I found the following:

Let $f,g$ be two continuous functions such that $f = g$ a.e. Then $f = g$ everywhere.

I want to know that if we replace $[a,b]$ by a general Lebesgue measurable set $E$, the same results holds.

Answer 1

It's false : let's consider $E=F=\{0,1\}$, $f:E\to F$ defined by $f(0)=f(1)=0$ and $g:E\to F$ defined by $g(0)=g(1)=1$. See that $E$ and $F$ are measurables and $f$ and $g$ are continuous, but $f=g$ a.e. and $f\neq g$.