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, and
fπ(μ) :=
A∈π:μ(A)>0
1A
ν(A)
μ(A)
, π ∈ P.
We recall that (fπ(μ))π∈P → dνa
dμ in L1(μ), as is (essentially) known. Here we
identify (πn)n∈N ⊆ P such that
fπn (μ) → dνa
dμ μ 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 (alreadyknown) characterisation of p-uniformly integrable martingales, without relying on
the martingale a.s. convergence theorem.