I am studying about Egorov's Theorem.
My teaching assistant said to me that Egorov's theorem is roughly like the following statement:
Under two conditions which are $|E|\lt+\infty$ and $|f|\lt+\infty$ , $$f_k \to f ~~\text{point-wisely} \Longrightarrow f_k \to f ~~\text{uniformly}$$
In the textbook, there is no word about point-wise. The definition of textbook version is like the following:
Egorov's Theorem. Suppose that $\{f_k\}$ is a sequence of measurable functions which converges almost everywhere in a set $E$ of finite measure to a finite limit $f$. Then given $\varepsilon\gt0$, there is a closed subset $F$ of $E$ such that $|E-F| \lt \varepsilon$ and $\{f_k\}$ converges uniformly to $f$ on $F$.
I am not sure that almost everywhere is considered as pointwise. Can someone tell me which are same or different? If they are different, please give a counter example. Although I know the definition of each of them, I cannot make example. :(