Proposition 2.3.50BB807E5CBD4

Proof. Let $u\in L^ p(\mu )$ and let $(K_ j)$ be a compact exhaustion of $\Omega $. The functions $u\mathbf1_{K_ j\cap \{ |u|\leq j\} }$ converge to $u$ in $L^ p(\mu )$ by dominated convergence (proposition 2.2.3), so it suffices to approximate a bounded Borel function $v$ vanishing outside a compact set $K$. A uniform approximation of $\operatorname{Re}v$ and $\operatorname{Im}v$ by simple functions vanishing outside $K$, with error $\delta $, has $L^ p(\mu )$ error at most $2\delta \mu (K)^{1/p}$; we are thus reduced to indicator functions $\mathbf1_ E$ of Borel sets $E\subset K$. Given $\varepsilon >0$, regularity of $\mu $ gives a compact set $K'\subset E$ and an open set $U\supset E$ with $\mu (U\setminus K')<\varepsilon $. A cutoff function $\eta $ for $K'$ in $U$ satisfies $|\eta -\mathbf1_ E|\leq \mathbf1_{U\setminus K'}$, so $\| \eta -\mathbf1_ E\| _{L^ p(\mu )}^ p<\varepsilon $. $\square$