Proof of the separability of $L^p (X,\mathcal{A},\mu)$ when $1 \leq p < \infty$ and $\mu$ is $\sigma$-finite.

Question

I took a course on Measure and Integration last semester and this semester on Functional Analysis. In the problem sets we were given the fact (we should know from last year's course) that

$L^p (X,\mathcal{A},\mu)$ is sepearable when $1 \leq p < \infty$ and $\mu$ is $\sigma$-finite.

Unfortunatly we did not prove this fact in the Measure and Integration course. Does anyone know the full proof of this? I would much appreciate if one could be detailed in proving it.

Thanks in advance!

Answer 1

Hint: The step functions are dense in $L^p(X,\mathcal{A},\mu)$ and $\mathbb{Q}$ is dense in $\mathbb{R}$ ( or $\mathbb{Q}+i\mathbb{Q}$ is dense in $\mathbb{C}$). Because of this and since $\mu$ is $\sigma$-finite,i.e. $X$ is the countable union of sets of finite measure, there is a countable and dense set of stepfunctions with values in $\mathbb{Q}$ (or $\mathbb{Q}+i\mathbb{Q}$).