  Let $\varphi$ be a locally upper bounded Borel measurable function on a
Greenian open set $\Omega$ in $R^d$ and, for every $x\in \Omega$, let
$v_\varphi(x)$ denote the infimum of the integrals of $\varphi$ with respect to
Jensen measures for $x$ on $\Omega$. Twenty years ago, B.J. Cole and T.J.
Ransford proved that $v_\varphi$ is the supremum of all subharmonic minorants
of $\varphi$ on $X$ and that the sets $\{v_\varphi<t\}$, $t\in R$, are
analytic. In this paper, a different method leading to the inf-sup-result
establishes at the same time that, in fact, $v_\varphi$ is the minimum of
$\varphi$ and a subharmonic function, and hence Borel measurable. This is
presented in the generality of harmonic spaces, where semipolar sets are polar,
and the key are measurability results for reduced functions on balayage spaces
which are of independent interest.
