Lemma 6

Let X be a locally compact topological space. Let $\mu$ be a positive measure on X and $p\in [1,+\infty [$. Let F be a finite subset of $L^p_K(X, \mu)$ and $\varepsilon >0$ a real number.

There exists a finite-rank projector $u$ of $L^p_K(X, \mu)$ of norm $\leqslant 1$ such that $\|u(f)-f\|\leqslant \varepsilon$ for every $f$ in F.

Let $\mathscr{P}$ be the set of finite partitions $\pi = (K_1, . . . ,K_n,H)$ of X, where $n\geqslant 1$ is an integer and $K_1, . . . ,K_n$ are integrable subsets of non-zero measure of X. For every partition $\pi = (K_1, . . . ,K_n,H)\in \mathscr{P}$, one defines an endomorphism $v_{\pi}$ of $\mathscr{L}^p(X, \mu)$ by putting

$i=1$

for $f\in \mathscr{L}_K^p(X, \mu)$, where $\varphi_{K_i}$ is the characteristic function of $K_i$. The mapping $v_{\pi}$ induces, on passing to quotients, a projector $u_{\pi}$ of $L^p_K(X, \mu)$. It is easily verified that the image of $u_{\pi}$ is the space of classes of functions $f\in \mathscr{L}_K^p(X, \mu)$ such that $f$ vanishes on H and is constant on $K_i$ for $1\leqslant i\leqslant n$.

Let us prove that $\|u_{\pi}\|\leqslant 1$. For $f\in \mathscr{L}_K^p(X, \mu)$, one has

By Hölder's inequality (INT, IV, p. 208, § 6, No.$^o4$, Theorem 2), for every $i$ one has the inequality

whence $\|u_{\pi}(f)\|_p\leqslant \|f\|_p$.

Let $\mathscr{E}$ be the set of classes in $L^p_K(X, \mu)$ of integrable functions on X taking only a finite number of values. Since $\mathscr{E}$ is dense in $L^p_K(X, \mu)$ (INT, IV, p. 162, §4, No.$^o10$, Corollary 1), there exists a finite set $F'$ of $\mathscr{E}$ such that every element of F is at distance at most $\varepsilon$ from an element of $F'$. On considering the finite partition $\pi$ formed by the sets of non-zero measure on which the mapping $x\mapsto (f(x))_{f\in F'}$ takes a given value, one sees that there exists an element $\pi$ of $\mathscr{P}$ such that $u_{\pi}(f) =f$ for every $f$ in $F'$. The projector $u_{\pi}$ therefore has the required properties.