### Pointwise compact and stable sets of measurable functions

by Fremlin and Shelah. [FrSh:406]

J Symbolic Logic, 1993

In a series of papers, M.Talagrand, the second author and
others investigated at length the properties and structure of
pointwise compact sets of measurable functions. A number of
problems, interesting in themselves and important for the theory of
Pettis integration, were solved subject to various special
axioms. It was left unclear just how far the special axioms were
necessary. In particular, several results depended on the fact that
it is consistent to suppose that every countable relatively
pointwise compact set of Lebesgue measurable functions is `stable'
in Talagrand's sense; the point being that stable sets are known to
have a variety of properties not shared by all pointwise compact
sets. In the present paper we present a model of set theory in which
there is a countable relatively pointwise compact set of Lebesgue
measurable functions which is not stable, and discuss the
significance of this model in relation to the original questions. A
feature of our model which may be of independent interest is the
following: in it, there is a closed negligible set Q subseteq
[0,1]^2 such that whenever D subseteq [0,1] has outer measure 1
then the set Q^{-1}[D]= {x:(exists y in D)((x,y) in Q)} has
inner measure 1.

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