Proof of existence of Schauder basis for $L^p(\Omega)$?

Question

There are a statements around, see [Brezis 2011, p. 146], like

All classical (separable) Banach spaces used in analysis have a Schauder basis .

I was wondering where to find a proof confirming this statement for $L^p(\Omega)$ with a domain $\Omega \subset \mathbb R^d$, $d\in\{2,3\}$, and with $1<p<\infty$.

In the book Bases in Banach Spaces by Singer, where also Brèzis references to, there is a proof for $L^p([0,1])$. Maybe, I just miss how to simply extend the arguments to higher dimensions.

Answer 1

One can easily construct measurable positive $\rho:\Omega\to\mathbb{R}_+$ such that $\int_\Omega\rho d\lambda_n=1$. Then consider a non-atomic measure $\nu=\rho^{1/p}d\lambda_n$ and its normalization $\mu=\nu(\Omega)^{-1}\nu$, then we have an isomorphism $$ I_1:L_p(\Omega,\lambda_n)\to L_p(\Omega,\mu):f\mapsto \rho^{-1/p}\cdot f $$ Thus $L_p(\Omega,\lambda_n)\cong L_p(\Omega,\mu)$ for some non-atomic probability measure $\mu$.

For any Polish space $K$ and non-atomic probability measure there exist measure-preserving Borel isomorphism $\sigma:\Omega\to[0,1]$ which indices isometric isomorphism $$ I_2:L_p(\Omega,\mu)\to L_p([0,1],\lambda_1):f\mapsto f\circ\sigma $$

The space $L_p([0,1],\lambda_1)$ for $p<\infty$ have a basis, for examplee the Haar system.

Thus we conclude $L_p(\Omega,\lambda_n)$ also have a basis.