  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.
