In the book Fourier Analysis and Self-adjointess of Reed and Simon in the proof of the
Feynman-Kac formular the author states that for any $V\in L^\infty (\mathbb{R}^3)$ there is a sequence $(V\_n)\_n$ of continuous and compact supported functions, which pointwise almost everywhere approximate $V$.


I do know some results concerning $C\_c(X)$ being dense in $L^p(X)$-spaces with respect to the $p$-norms. This holds only if $p<\infty$ (Rudin Real and Complexanalysis). Of course we are talking about a different topology. But I cannot find a good explanation why the property I stated holds. Actually to make it more precise, but probably an easy modification, the author wants the sequence to be uniformly bounded by $\|V\|\_\infty$.



Any $L^\infty (\mathbb{R}^3)$ can be approximated pointwise almost everywhere by continuous function with compact support
Let $V\in L^\infty$ and let $V\_\varepsilon=V\*\varphi\_\varepsilon$ be a standard approximation by convolution. Since $V\in L^1$ on any bounded set, it follows that $V\_\varepsilon\to V$ in $L^1$ on bounded sets. Since a convergence in $L^1$ implies a.e. convergence on a subsequence, for every bounded set you find a subsequence that converges a.e. Then using a standard diagonal argument you find a subsequence that converges a.e. in $\mathbb{R}^3$.


