Lemma 2

— Let $\alpha$ be a bounded positive measure on $T$, and $\beta$ a real measure on $T$ such that $|\beta| \leq M \alpha$, where $M$ is a positive constant. Then, there exists a real function $u$, $\alpha$-integrable, such that $\beta = u \cdot \alpha$.

Let $g$ be an element of the space $\mathscr{L}_\mathbf{R}^2(T, \alpha)$; $g$ is $\beta$-measurable and $\int^\bullet |g|^2 \, d|\beta| \leq M \int^\bullet |g|^2 \, d\alpha < +\infty$. The function $g$ therefore belongs to $\mathscr{L}^2(T, |\beta|)$, and also to $\mathscr{L}^1(T, |\beta|)$ since $\beta$ is bounded. By the Cauchy–Schwarz inequality,

The mapping $g \mapsto \beta(g)$ is thus a continuous linear form on $\mathscr{L}^2(T, \alpha)$. The Hausdorff space associated with $\mathscr{L}^2(T, \alpha)$ being a Hilbert space, there then exists (TVS, Ch. V, § 1, No. 7, Th. 3) a real function $u \in \mathscr{L}^2(T, \alpha)$, therefore also belonging to $\mathscr{L}^1(T, \alpha)$, such that $\beta(g) = \alpha(ug)$ for every $g \in \mathscr{L}^2(T, \alpha)$. Applying this relation for $g \in \mathscr{K}(T)$, one sees that $\beta = u \cdot \alpha$.