  This paper is on developing stochastic analysis simultaneously under a
general family of probability measures that are not dominated by a single
probability measure. The interest in this question originates from the
probabilistic representations of fully nonlinear partial differential equations
and applications to mathematical finance. The existing literature relies either
on the capacity theory (by Denis and Martini), or on the underlying nonlinear
partial differential equation (by Peng). In both approaches, the resulting
theory requires the smoothness of the corresponding processes and random
variables in terms of the underlying canonical process. In this paper, we
investigate this question for a larger class of "non-smooth" processes, but
with a restricted family of non-dominated probability measures. For smooth
processes, our approach leads to similar results as in previous literature,
provided the restricted family satisfies an additional density property.
