  We investigate Keisler measures in arbitrary theories. Our initial focus is
on Borel definability. We show that when working over countable parameter sets
in countable theories, Borel definable measures are closed under Morley
products and satisfy associativity. However, we also demonstrate failures of
both properties over uncountable parameter sets. In particular, we show that
the Morley product of Borel definable types need not be Borel definable
(correcting an erroneous result from the literature). We then study various
notions of generic stability for Keisler measures and generalize several
results from the NIP setting to arbitrary theories. We also prove some positive
results for the class of frequency interpretation measures in arbitrary
theories, namely, that such measures are closed under convex combinations and
commute with all Borel definable measures. Finally, we construct the first
example of a complete type which is definable and finitely satisfiable in a
small model, but not finitely approximated over any small model.
