  We consider, and make precise, a certain extension of the Radon-Nikodym
derivative operator, to functions which are additive, but not necessarily
sigma-additive, on a subset of a given sigma-algebra. We give applications to
probability theory; in particular, to the study of $\mu$-Brownian motion, to
stochastic calculus via generalized It\^o-integrals, and their adjoints (in the
form of generalized stochastic derivatives), to systems of transition
probability operators indexed by families of measures $\mu$, and to adjoints of
composition operators.

Probability