Differentiation of measures on an arbitrary measurable space

Abstract

Let μ,ν be positive finite measures on an arbitrary measurable space (Ω,F), ν=νa+νs be the Lebesgue decomposition of ν with respect to μ, P be the family of all finite partitions π⊆F of Ω, andfπ(μ):=∑A∈π:μ(A)>01Aν(A)μ(A),π∈P. We recall that (fπ(μ))π∈P→dνadμ in L1(μ), as is (essentially) known. Here we identify(πn)n∈N⊆P such thatfπn(μ)→dνadμμ a.s. as n→∞; in the setting of separable F, a (rather trivial) way to do this was already known. To do all this, we characterise the case of equality in Jensen's conditional inequality (generalising the known case of the standard Jensen inequality), and use this to determine how, given a p-uniformly integrable martingale (fi)i∈I, one can identify a sequence(in)n∈N⊆I such that (fin)n converges in Lp to some f which closes the whole net(fi)i. We also give a new proof of the (already-known) characterisation of p-uniformly integrable martingales, without relying on the martingale a.s. convergence theorem.