$f, g$ are continuous functions on $E$ where $E$ is measurable set. If $f=g$ a.e on $E$ then $f=g$ on $E$?

Question

problem

$f, g$ are continuous functions on $E$ where $E$ is Lebesgue measurable set. If $f=g$ a.e on $E$ then $f=g$ on $E$? If not, give a counterexample.

effort

Let us change $[a,b]$ with $\{0\}$ which is a measurable set. Let $f: \{0\} \rightarrow \mathbb{R}$ and $g: \{0\} \rightarrow \mathbb{R}$ be functions. Then, $f$ and $g$ are continuous, and clearly $f=g$ a.e. But of course $f$ must not be equal to $g$ necessarily.

question

1. Is the example proper?
2. Could you tell me more details?

refer

Show that if $f=g \text{a.e}$ on $[a,b]$ implies that $f=g$ on $[a,b]$.

Thanks!