$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a probability density function such that $\rho_n \in C_c^{\infty} (\bR^d)$ and $\supp \rho_n \subset \overline{B(0,1 / n)}$. We have from Brezis' Functional Anlysis that

Theorem 4.22. Assume $f \in L^p (\bR^d)$ with $1 \le p<\infty$. Then $\| \rho_n * f -f \|_{L^p} \to 0$ as $n\to \infty$.

Above, $*$ denotes the mollification operation. Now we fix a Borel probability measure $\mu$ on $\bR^d$. Is the following statement true?

Assume $f \in L^p (\bR^d, \mu)$ with $1 \le p<\infty$. Then $\| \rho_n * f -f \|_{L^p (\mu)} \to 0$ as $n\to \infty$.

Any reference is appreciated. Thank you for your elaboration.

$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\supp}{\operatorname{supp}} $ Let $(\rho_n)_{n \geq 1}$ be a sequence of mollifiers on $\bR^d$, i.e., each $\rho_n$ is a probability density function such that $\rho_n \in C_c^{\infty} (\bR^d)$ and $\supp \rho_n \subset \overline{B(0,1 / n)}$. We have from Brezis' Functional Anlysis that

Theorem 4.22. Assume $f \in L^p (\bR^d)$ with $1 \le p<\infty$. Then $\| \rho_n * f -f \|_{L^p} \to 0$ as $n\to \infty$.

Above, $*$ denotes the mollification operation. Now we fix a Borel probability measure $\mu$ on $\bR^d$. Is the following statement true?

Assume $f \in L^p (\bR^d, \mu)$ with $1 \le p<\infty$. Then $\| \rho_n * f -f \|_{L^p (\mu)} \to 0$ as $n\to \infty$.

References are appreciated. Thank you for your elaboration.