  The analogy between determinantal point processes (DPPs) and free fermionic
calculi is well-known. We point out that, from the perspective of free
fermionic algebras, Pfaffian point processes (PfPPs) naturally emerge, and show
that a positive contraction acting on a "doubled" one-particle space with an
additional structure defines a unique PfPP. Recently, Olshanski inverted the
direction from free fermions to DPPs, proposed a scheme to construct a
fermionic state from a quasi-invariant probability measure, and introduced the
notion of perfectness of a probability measure. We propose a method to check
the perfectness and show that Schur measures are perfect as long as they are
quasi-invariant under the action of the symmetric group. We also study
conditional measures for PfPPs associated with projection operators.
Consequently, we show that the conditional measures are again PfPPs associated
with projection operators onto subspaces explicitly described.
