  A general theory of resource-bounded measurability and measure is developed.
Starting from any feasible probability measure $\nu$ on the Cantor space $\C$
and any suitable complexity class $C \subseteq \C$, the theory identifies the
subsets of $\C$ that are $\nu$-measurable in $C$ and assigns measures to these
sets, thereby endowing $C$ with internal measure-theoretic structure. Classes
to which the theory applies include various exponential time and space
complexity classes, the class of all decidable languages, and the Cantor space
itself, on which the resource-bounded theory is shown to agree with the
classical theory.
  The sets that are $\nu$-measurable in $C$ are shown to form an algebra
relative to which $\nu$-measure is well-behaved. This algebra is also shown to
be complete and closed under sufficiently uniform infinitary unions and
intersections, and $\nu$-measure in $C$ is shown to have the appropriate
additivity and monotone convergence properties with respect to such infinitary
operations.
  A generalization of the classical Kolmogorov zero-one law is proven, showing
that when $\nu$ is any feasible coin-toss probability measure on $\C$, every
set that is $\nu$-measurable in $C$ and (like most complexity classes)
invariant under finite alterations must have $\nu$-measure 0 or $\nu$-measure 1
in $C$.
  The theory is presented here is based on resource-bounded martingale
splitting operators, which are type-2 functionals, each of which maps $\N
\times {\cal D}_\nu$ into ${\cal D}_\nu \times {\cal D}_\nu$, where ${\cal
D}_\nu$ is the set of all $\nu$-martingales. This type-2 aspect of the theory
appears to be essential for general $\nu$-measure in complexity classes $C$,
but the sets of $\nu$-measure 0 or 1 in C are shown to be characterized by the
success conditions for martingales (type-1 functions) that have been used in
resource-bounded measure to date.
