Approximation of function in general measure space

Question

Let $\mu$ be a $\sigma$-finite measure on $R^n$ ($n\geq 1$) and $(E,d)$ be a complete metric space. For any measurable function $f: R^n\to E$ with $$ \int_{R^n}d(f(x),f(x_0))\mu(dx)<\infty,\quad \forall x_0\in R^n, $$ is there a sequence of continuous function $f_n$ such that

$$ \lim_{n\to\infty}\int_{R^n}d(f_n(x),f(x))\mu(dx)=0 \quad?? $$ Is there any reference for this?

Answer 1

No, let $\mu$ be Lebesgue measure on $[0,1] \subset \mathbb{R}$ (and zero outside $[0,1]$), let $E = \{0,1\}$, and let $f(t) = \begin{cases}0&t < 1/2\cr 1& t\geq 1/2\end{cases}$. The only continuous functions are two constant functions, so the kind of approximation you want is impossible.