  Random sets are used to get a continuous partition of the cardinality of the
union of many overlapping sets. The formalism uses M\"obius transforms and
adapts Shapley's methodology in cooperative game theory, into the context of
set theory. These ideas are subsequently generalized into the context of
finite-dimensional Hilbert spaces. Using random projectors into the subspaces
spanned by states from a total set, we construct an infinite number of
continuous resolutions of the identity, that involve Hermitian positive
semi-definite operators. The simplest one is the diagonal continuous resolution
of the identity, and it is used to expand an arbitrary vector in terms of a
continuum of components. It is also used to define the $F(x_1,x_2)$ function on
the `probabilistic quadrant' $[0,\infty) \times [0,\infty)$, which is analogous
to the Wigner function for the harmonic oscillator, on the phase-space plane.
Systems with finite-dimensional Hilbert space (which are naturally described
with discrete variables) are described here with continuous probabilistic
variables.
