Reduced functions and Jensen measures

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.