  For a Gaussian measure on a separable Hilbert space with covariance operator
$C$, we show that the family of conditional measures associated with
conditioning on a closed subspace $S^{\perp}$ are Gaussian with covariance
operator the short $\mathcal{S}(C)$ of the operator $C$ to $S$. We provide two
proofs. The first uses the theory of Gaussian Hilbert spaces and a
characterization of the shorted operator by Andersen and Trapp. The second uses
recent developments by Corach, Maestripieri and Stojanoff on the relationship
between the shorted operator and $C$-symmetric oblique projections onto
$S^{\perp}$. To obtain the assertion when such projections do not exist, we
develop an approximation result for the shorted operator by showing, for any
positive operator $A$, how to construct a sequence of approximating operators
$A^{n}$ which possess $A^{n}$-symmetric oblique projections onto $S^{\perp}$
such that the sequence of shorted operators $\mathcal{S}(A^{n})$ converges to
$\mathcal{S}(A)$ in the weak operator topology. This result combined with the
martingale convergence of random variables associated with the corresponding
approximations $C^{n}$ establishes the main assertion in general. Moreover, it
in turn strengthens the approximation theorem for shorted operator when the
operator is trace class; then the sequence of shorted operators
$\mathcal{S}(A^{n})$ converges to $\mathcal{S}(A)$ in trace norm.
