We say that the condition of commutation is satisfied if the
conditional expectation operators E(IFS ) and E(-.F ) commute,
s t
i.e., if E(.IF1F ) = E(.IF IF ). The product is then equal to
E(-IF1 nF2) = E(-IFt). (In fact, denote, for f c L (P), E a Banach
space with the RNP, g = E(flF IF2) = E(fF2 IF1). Then g is measurable
2 1
with respect to F2 and Fs, hence g is F t-measurable. Also, for
t s st
1 2
A E Fst, we have IAgdP = JAE(f Fsl F)dP = AE(fFs)dP = JAfdP
since A E F = F1 (1 2 Thus, g = E(fIF ).) All of the main
st s t St
results of the theory of two-parameter processes require the condition
of commutation.
The second (and more frequently used) way of describing
filtrations on R2 is due to Cairoli and Walsh [2]. There, we begin
with a family (Fz, z c R2} of sub-a-fields of F satisfying the
following axioms:
(Fl) If ziz', then F C F' (this is (2) of Meyer).
z z
(F2) F contains all the negligible sets of F. (Note: This,
o
along with (F1), implies condition (1) of the Meyer
construction.)
(F3) For each z, F = ( F .. (This was condition (3) of
z z'>z z
z'>z
Meyer.)
We now define F = F = V Fst F2 F = V Fst. In place of
t s
the condition of commutation, we impose the axiom
(F4) For each z, F and F are conditionally independent given
z z
F, i.e., for Y F1-measurable, integrable, Y2 F2
measurable, integrable, we have