Integrating $\chi_\mathbb{Q}e^{\|x|}$ over counting measure.

Question

My solution: Since integrating over counting measure is just summation we just have to consider $\sum_{q \in Q}F(q)$ However, this sum is larger than $\sum_{n \in \mathbb{N}}F(n)=\sum_{n\in \mathbb{N}}e^{-1/n}=\infty$. Thus $F$ is no integrable.

b) Is true as integrable simple functions are always dense in $L^P$ spaces.

Answer 1

This all looks good to me. Note that integrable simple functions are dense in $L^p$ for $p<\infty$, which of course applies here.

Answer 2

Your answer for b) is not clear enough. Simple functions take only finitely many values but they need not be finitely supported. If $f$ is integrable then $\{x: f(x) \neq 0\}$ is at most countable. Suppose $f(x)=0$ whenever $x \notin \{x_1,x_2,...\}$. Define $f_n(x)=0$ if $x \notin \{x_1,x_2,...,x_n\}$ and $f_n(x)=f(x)$ otherwise. THen $f_n$'s are fintely supported and converge to $f$ in $L^{1}$.