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