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

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

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?