  The finite Hilbert transform $T$ is a classical (singular) kernel operator
which is continuous in every rearrangement invariant space $X$ over $(-1,1)$
having non-trivial Boyd indices. For $X=L^p$, $1<p<\infty$, this operator has
been intensively investigated since the 1940's (also under the guise of the
``airfoil equation''). Recently, the extension and inversion of $T\colon X\to
X$ for more general $X$ has been studied in G. P. Curbera, S. Okada, W. J.
Ricker, Inversion and extension of the finite Hilbert transform on $(-1,1)$,
Ann. Mat. Pura Appl. 198 (2019), 1835-1860, where it is shown that there exists
a larger space $[T,X]$, optimal in a well defined sense, which contains $X$
continuously and such that $T$ can be extended to a continuous linear operator
$T\colon[T,X]\to X$. The purpose of this paper is to continue this
investigation of $T$ via a consideration of the $X$-valued vector measure
$m_X\colon A\mapsto T(\chi_A)$ induced by $T$ and its associated integration
operator $f\mapsto \int_{-1}^1f\,dm_X$. In particular, we present integral
representations of $T\colon X\to X$ based on the $L^1$-space of $m_X$ and other
related spaces of integrable functions.

Functional Analysis