I was wondering if $L^p(\mathbb{R}^n)$ is separable when $\mathbb{R}^n$ is Lebesgue measure space. I know $L^p(X)$ is separable, provided $X$ is a separable measure space. Consequently, since $\mathbb{R}^n$ with Borel measure is a separable measure space, $L^p(\mathbb{R}^n)$ is separable if $\mathbb{R}^n$ is Borel measure space. It seems that $L^p(\mathbb{R}^n)$ is not separable over Lebesgue measure space. Is it true?
Thanks in advance.