Exercise 19, § 1

Let X be a locally compact, metrizable and countable at infinity topological space and $\mu$ a positive measure on X. Denote by A the set of atoms of $\mu$. Let $f\in \mathscr{L}^{\infty}(X, \mu)$ and $p\in [1,+\infty ]$.

Prove that multiplication by $f$ defines a compact endomorphism of $L^p(X, \mu)$ if, and only if, the following two conditions are satisfied:

(i) The set $\{x\in X$ - $A|f(x)\not = 0\}$ is $\mu$-negligible ;

(ii) For every $\varepsilon >0$, there exists a finite subset F of A such that $|f(a)|\leqslant \varepsilon$ for all $a\in A$ - F.

(Consider multiplication by $f$ on the space $L^p(X_{\varepsilon}, \mu)$, where $X_{\varepsilon}$ is the set of the $x\in X$ such that $|f(x)|\geqslant \varepsilon .$)