  We develop a theory of generically stable and smooth Keisler measures in NIP
metric theories, generalizing the case of classical logic. Using smooth
extensions, we verify that fundamental properties of (Borel)-definable measures
and the Morley product hold in the NIP metric setting. With these results, we
prove that as in discrete logic, generic stability can be defined equivalently
through definability properties, statistical properties, or behavior under the
Morley product. We also examine weakly orthogonal Keisler measures,
characterizing weak orthogonality in terms of various analytic regularity
properties.
  We then examine Keisler measures in distal metric theories, proving that as
in discrete logic, distality is characterized by all generically stable
measures being smooth, or by all pairs of generically stable measures being
weakly orthogonal. We then use this, together with our results on weak
orthogonality and a cutting lemma, to find analytic versions of distal
regularity and the strong Erd\H{o}s-Hajnal property.
