Measure theoretic aspects of the finite Hilbert transform

The finite Hilbert transform $T$, when acting in the classical Zygmund space $\text{logl}$ (over $(-1,1)$), was intensively studied in \cite{curbera-okada-ricker-log}. In this note an integral representation of $T$ is established via the $L^1(-1,1)$-valued measure $\text{mlog}\text{\hspace{0.17em}⁣}:A\mapsto T({\chi }_A)$ for each Borel set $A\subseteq (-1,1)$. This integral representation, together with various non-trivial properties of $\text{mlog}$, allow the use of measure theoretic methods (not available in \cite{curbera-okada-ricker-log}) to establish new properties of $T$. For instance, as an operator between Banach function spaces $T$ is not order bounded, it is not completely continuous and neither is it weakly compact. An appropriate Parseval formula for $T$ plays a crucial role.