Space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable

Question

Let's consider the space $\mathcal{L}^p(X, \Sigma, \mu)$ of all functions $f\colon X \to \mathbb{R}$ (or $\mathbb{C}$) for which: $$ \int\limits_X|f|^p \mu(dx) < \infty. $$ Here $X$ is a metric space, $\Sigma\subset 2^X$ — sigma-algebra on $X$ and $\mu$ — is the measure on $\Sigma$. As usual, functions that are equal almost everywhere considered to be equivalent, so technically we are dealing with classes $$[f]=\{g\in \mathcal{L}^\infty(X,\Sigma, \mu) \colon \mu \{x\in X\colon f(x)\neq g(x)\}=0\}.$$ This space is endowed with integral metric $$\rho_p(f,g)=\left(\int\limits_X|f-g|^p \mu(dx)\right)^{1/p}$$

The statement is that space $\mathcal{L}^p(X, \Sigma, \mu)$ is separable iff $(\Sigma, \rho_\Delta)$ is separable, where $\forall A,B\in\Sigma$ $$\rho_\Delta(A,B)=\mu(A\Delta B)$$ Here, again, we consider $A\in \Sigma$ as a class $[A]=\{A'\in\Sigma\colon \mu(A\Delta A')=0\}$ to have the identity of indiscernibles.

How do one prove the statement?

Answer 1

As it has been pointed out by Ramiro, the assertion to be proved in the question is false. Only one of the implications is correct that is if $(\Sigma, \rho_{\Delta})$ is separable, then $L^p(X,\Sigma, \mu)$ is separable. This is what will be shown below.

Let $\chi_A$ be the characteristic function of the set $A$. Now observe that

$\| \chi_A - \chi_B\|_{L^p}^p = \int|\chi_A - \chi_B|^p = \int_{A\setminus B}1 + \int_{B\setminus A}1 = \mu(A\setminus B) + \mu(B\setminus A) = \rho_{\triangle}(A,B)$

Now consider the function $\varphi: \Sigma \to L^p(X, \Sigma,\mu )$ defined as $ A \mapsto \chi_A$.

This is an injective map from $(\Sigma, \rho_{\triangle})$ to $(L^p(X, \Sigma,\mu), \rho_p)$.

Observe that $\rho_{\triangle}(A_i,B) \rightarrow 0 \Leftrightarrow \|\chi_{A_i} - \chi_{B} \|_{L^p} \rightarrow 0$.

Consider a countable dense subset in $L^p$ say $f_i$. Now choose an element from each of the following sets $B(f_i,1/k)\cap \mathrm{Im}(\varphi)$ say $\chi_{A_{i,k}}$ where $i$ and $k$ vary over natural numbers. This will be a countable subset of $\mathrm{Im}(\varphi)$. For any given $\varepsilon$ and $\chi_A$ there exists $f_j$ such that $\| f_j - \chi_A \|_{L^p} < 1/k < \varepsilon/2$. Now let's choose $\chi_{A_{j,2k}}$, then $\| \chi_{A_{j,2k}} - \chi_A \|_{L^p} \leq \varepsilon $. Mistake in this argument was also pointed out by Glinka below.

Let $\lbrace A_i \rbrace$ to be a dense subset in $\Sigma$. Now consider the vector space spanned by $\mathbb{Q}\chi_{A_i}$. This is a countable set. This is dense in the vector space spanned by $\mathbb{R}\chi_{A_i}$ which is dense in $L^p$ since simple functions are dense in $L^p$. One can suitably modify the proof for complex valued functions.

Answer 2

The correct statement is that the space of summable functions (modulo a.e. equivalence) is separable if and only if there is a countable collection C of measurable sets of finite measure such that for each measurable set Q of FINITE measure and each integer N there is a set A in the collection C such that the measure of the symmetric difference between Q and A is less that 1/N. I have been unable to find a published proof of this statement, but the assertion is made in a book by Zaanen on measure theory published around 1967. However, the proof of the tricky direction made there is wrong. I know how to prove this result. If readers are interested I will provide a proof here.