  We introduce the notion of an ACF space, that is, a space for which a
generalized version of M. Riesz's theorem for conjugate functions with values
in the Banach space is bounded. We use transference to prove that spaces for
which the Hilbert transform is bounded, i\.e\. $X\in\text{HT}$, are ACF spaces.
We then show that Bourgain's proof of $X\in\text{HT}\implies X\in\text{UMD}$ is
a consequence of this result.
