Proposition 1

Let X be a locally compact topological space. Let $\mu$ be a positive measure on X and let $p\in [1,+\infty [$. Let $(X_i)_{i\in I}$ be a locally countable family (INT, IV, p. 190, § 5, n$^o9$, def. 7) of pairwise disjoint locally compact subsets of X such that the complement of the union of the $X_i$ is locally $\mu$-negligible. Let $\varphi_i$ be the characteristic function of $X_i$ and $\mu_i$ the measure induced by $\mu$ on $X_i$ (INT, IV, p. 186, §5, n$^o7$, def. 4).

a) The mapping $p_i:f\mapsto f \varphi_i$ is a projector of $\mathscr{L}^p(X, \mu)$ and defines by passing to the quotients a projector $\widetilde{p}_i$ of $L^p(X, \mu)$. The latter is an orthoprojector of $L^2(X, \mu)$ if $p= 2$;

b) The mapping $f\mapsto f|X_i$ induces an isometric isomorphism of the image of $p_i$ onto the space $\mathscr{L}^p(X_i, \mu_i)$ and defines by passing to the quotients an isometric isomorphism of the image of $\widetilde{p}_i$ onto $L^p(X_i, \mu_i)$, through which these two spaces are identified;

c) The sum of the spaces $L^p(X_i, \mu_i)$ is total in the space $L^p(X, \mu)$. In the case $p= 2$, the space $L^2(X, \mu)$ is the Hilbert sum of the spaces $L^2(X_i, \mu_i)$.

Assertion a) is elementary. Assertion b) results from the scholium of INT, V, p. 84, § 7, n$^o1$.

Let $q\in ]1,+\infty ]$ be the conjugate exponent of $p$. We identify the dual of $L^p(X, \mu)$ with $L^q(X, \mu)$ (INT, V, p. 61, § 5, n$^o8$, th. 4). Let $f$ be a function in $\mathscr{L}^q(X, \mu)$ whose class $\widetilde{f}$ in $L^q(X, \mu)$ satisfies $\langle \widetilde{f} ,\widetilde{p}_i(\varphi )\rangle = 0$ for all $i\in I$ and all $\varphi \in L^p(X, \mu)$. Since the image of $p_i$ contains $\mathscr{K}(X_i)$, it follows that the measure $(f|X_i)\cdot \mu_i$ on $X_i$ is zero, which means that the restriction of $f$ to $X_i$ is locally $\mu_i$-negligible on $X_i$ (INT, V, p. 46, § 5, n$^o3$, cor. 2). Since the complement in X of the union of the $X_i$ is locally $\mu$-negligible, we conclude that the function $f$ is locally $\mu$-negligible on X (INT, IV, p. 190, § 5, n$^o9$ and p. 172, n$^o2$, prop. 5). The class of $f$ is then zero in $L^q(X, \mu)$ (using INT, V, p. 8, § 1, n$^o3$, lemma 1 and the corollary of proposition 9 when $p\not = 1$). By the Hahn–Banach theorem (EVT, II, p. 46, cor. 1), the first part of the assertion c) follows.

If $p= 2$, the image of $p_i$ is orthogonal to that of $p_j$ for all $i\not =j$, since $X_i$ and $X_j$ are then disjoint, whence the last assertion.