Exercise 18, § 2

Let X be a compact metrisable space and let $\mu$ be a positive measure on X. Let us denote by $E = L^2_{\mathbf{C}}(X, \mu)$. Let $(\mathscr{X}_n)_{n\in\mathbf{N}}$ be a sequence of finite partitions of X by measurable sets such that $\mathscr{X}_{n+1}$ is finer than $\mathscr{X}_n$ for every $n\in \mathbf{N}$, and such that the largest of the diameters of a subset of $\mathscr{X}_n$ tends to 0 as $n\rightarrow +\infty$.

Let $E_n\subset E$ be the space of the $f\in L^2(X, \mu)$ such that $f$ is $\mu$-almost everywhere constant on Y for every subset Y of $\mathscr{X}_n$. Put $E_{-1}=\{0\}$. For every $n\geqslant -1$, let $p_n$ be the orthogonal projector on E with image $E_n$.

a) The sequence $(p_n)_{n\in\mathbf{N}}$ converges to the identity in the space $\mathscr{L}_s(E)$ of endomorphisms of E endowed with the topology of simple convergence. (First consider the limit of $p_n(f)$ when $f\in \mathscr{C}(X).$)

b) For every $n\in \mathbf{N}$, the endomorphism $q_n=p_n-p_{n-1}$ is the orthogonal projector on $F_n= E_n\cap E^{\circ}_{n-1}$, and the spaces $F_n$ are pairwise orthogonal.

c) One has $\sum_nq_n= 1_E$ in $\mathscr{L}_s$(E), and the space E is the Hilbert sum of the spaces $F_n$.