If either $f_n \longrightarrow f $ almost everywhere OR $f_n \longrightarrow f$

Question

[1] Are all simple functions Lebesgue Integrable? Prove or disprove.

[2] If either $f_n \longrightarrow f $ almost everywhere OR $f_n \longrightarrow f$ in measure then show that f is finite valued (ie real valued) almost everywhere.

MY ATTEMPT

[1] I'm not sure if all simple functions are Lebesgue integrable or not but this is my example of a simple function which is not Lebesgue integrable

Let $(\mathbb{R},B_\mathbb{R},\mu_L)$ be a Lebesgue measure space and

$\phi:[0,\infty] \longrightarrow \mathbb{R}$ is a simple function defined as

$$ \phi(x)= \begin{cases} -1, if x \in \bigcup_{x \in \mathbb{Z^+}} [2k+1,2k+2],\\ 1, if x \in \bigcup_{x \in \mathbb{Z^+}} [2k,2k+1] \end{cases} $$

[2] Not sure how to start or which one to pick.In either case i'm not sure if I have to take some sub sequence and then show it is finite and if so what's my logic and steps?

Answer 1

[1] Think of a constant.

[2] For the a.e. convergence, remember a convergent sequence is bounded. For the convergence in measure remember there is a subsequence converging a.e.