  We investigate the behavior of 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, closure under Morley products (correcting
another erroneous result from the literature) and convex combinations. 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.
