Subharmonicity without Upper Semicontinuity

LetΩbe an open subset ofRd(d⩾2). Givenx∈Ω, a Jensenmeasureforxis a Borel probability measureμ, supported on a compact subset ofΩ, such that each subharmonic functionuonΩsatisfiesu(x)⩽∫ u dμ. ((*))A functionuonΩ(subject to certain measurability conditions, much weaker for example than upper semicontinuity) is calledquasi-subharmonicif it satisfies (*) for eachx∈Ωand each Jensen measureμforx. Our first result is thatuis quasi-subharmonic onΩif and only if its upper semicontinuous regularizationu* is subharmonic onΩand equal tououtside a set of capacity zero. This easily implies, for example, Cartanʹs classical theorem on the supremum of a locally bounded family of subharmonic functions. Our second result is a converse to Cartanʹs theorem: provided thatΩhas a Greenʹs function, every quasi-subharmonic function 4 onΩis the supremum of some family of subharmonic functions. These ideas eventually lead to the following duality theorem. Ifϕis a Borel function locally bounded above onΩ, and ifΩhas a Greenʹs function, then for each x∈Ω,sup{v(x): vsubharmonic on Ω v⩽ϕ}=inf ∫ ϕ dμ: μ is a Jensen measure for x.We also investigate to what extent these results carry over to plurisubharmonic functions.