Suppose that $f_n(x)$ converges pointwise on $I$, i.e., for every fixed $x \in I$, we have $f_n(x) \to l_x$. Then, there exists a correspondence $x \in I \to \lim_{n \to \infty} f_n(x) = l_x \in \mathbb{R}$. This correspondence is a function due to the uniqueness theorem of limits. We demonstrate that this function is defined as $f: x \in I \to l_x \in \mathbb{R}$, with $l_x = f(x)$. This function is called the **limit function**.
**Definition**: $f_n$ converges pointwise to $f$ on $I$ if and only if:
Suppose that $(f_n)_n$ is a sequence of measurable functions on a set $E$ and that $f_n \to f$ a.e.on $E$. Does this imply that $f$ is measurable?
I know that pointwise limit of measurable function is measurable. But here we only have convergence a.e. So I got confused.
