Small $\ell^p$ spaces are obtainable from $L^p$

Question

I've seen that in a lot of books there is written that $$l^p=L^p(X,\Sigma,\mu),$$ where $X=\Bbb N, \Sigma=P(\Bbb N), \mu=\#$, ($\#$ is the counting measure). I would like to see how to prove it, because I've tried but there are a lot of things that don't convince me. Do you know where I can find this proved, or some suggestion to prove it?

Answer 1

Here is an outline for proving your statement.

1. $\ell^p$ is a space of (some) functions $\mathbb N\to\mathbb R$, and so is $L^p$ (with the choices indicated by OP). (I chose to work with the reals, but this choice is irrelevant.) The counting measure gives nonzero measure to all nonempty sets, so each equivalence class in $L^p$ only consists of one function. (As Nate Eldredge correctly reminded me, the $L^p$ space is a space of equivalence classes of functions rather than a space of functions. But now every function is equivalent to itself only, so we can regard $L^p$ as an actual space of functions.)
2. Every set is measurable with respect to the counting measure, so all functions $\mathbb N\to\mathbb R$ are measurable.
3. It remains to show that a function has finite $\ell^p$ norm if and only if it has finite $L^p$ norm.
4. Let $f:\mathbb N\to\mathbb R$. Then $$ \|f\|_{\ell^p}^p = \sum_{n=1}^\infty|f(n)|^p = \int_{\mathbb N}|f(x)|^pd\#(x) = \|f\|_{L^p}^p. $$

If there are things that you are not convinced about, what are they? I numbered the steps to make them easier to refer to.