Let phi be a locally upper bounded Borel measurable function on a Greenian open set Omega in R-d and, for every x is an element of Omega, let v(phi)( x) denote the infimum of the integrals of phi with respect to Jensen measures for x on Omega. Twenty years ago, B.J.

Cole and T.J. Ransford proved that v(phi) is the supremum of all subharmonic minorants of phi on X and that the sets {v(phi) < t}, t is an element of R, are analytic.

In this paper, a different method leading to the inf-supresult establishes at the same time that, in fact, v(phi) is the minimum of phi 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 tools are measurability results for reduced functions on balayage spaces which are of independent interest.