Hilbert's transformation and $A$-integral

We prove that if $g$ is a bounded function, $g\in L^p(\mathbb R)$, $p\ge1$, its Hilbert's transformation $\tilde g$ is also a bounded function, and $f(x)\in L(\mathbb R)$, then $\tilde fg$ is an $A$-integrable function on $\mathbb R$ and
$$ (A)\!\int\limits_{\mathbb R}\tilde fg\,dx =-(L)\!\int\limits_{\mathbb R}f\tilde g\,dx. $$