Course in Probability Theory

9 .1 BASIC PROPERTIES OF CONDITIONAL EXPECTATION 1 3 1 5Hence by (6),X(B) = / Yd°J'= cp(x)dµ)JA 1 BThis being true for every B in 73 1 , it follows that cp is a version of the derivativedX/dµ . Theorem 9 .1 .2 is proved .IAs a consequence of the theorem, the function u(Y IX) of w is constanta .e . on each set on which X (w) is constant . By an abuse of notation, the co(x)above is sometimes written as S°(Y X = x) . We may then write, for example,for each real c :Y dGJ' = ~(Y I X = x) dGJ'{X < x} .{X_4 :501-00,C]Generalization to a finite number of X's is straightforward . Thus oneversion of (9(Y I X 1 , . . ., X,) is cp(X 1 , . . ., X,), where co is an n-dimensionalBorel measurable function, and by m`(Y I X 1 = xl , . . . , X, = x n ) is meant(p(xl, , xn)-It is worthwhile to point out the extreme cases of ~, (Y I;6?) :°(Y if) = e(Y), m I ~) = Y ; a .e .where / is the trivial field {0, Q} . If ' is the field generated by one setA : {P , A, A', S2}, then e(Y I -6-7) is equal to &(Y I A) on A and e(Y I A c )on A` . All these equations, as hereafter, are between equivalent classes ofr .v .'s .We shall suppose the pair ( .97, -OP) to be complete and each Borel subfieldof JF to be augmented (see Exercise 20 of Sec . 2 .2) . But even if isnot augmented and ~ is its augmentation, it follows from the definition thatdt(Y I ) = i (Y I ), since an r .v . belonging to 3~ is equal to one belongingto ;6~ almost everywhere (why ?) . Finally, if ~j-o is a field generating a7 , or justa collection of sets whose finite disjoint unions form such a field, then thevalidity of (6) for each A in -fo is sufficient for (6) as it stands . This followseasily from Theorem 2.2 .3 .The next result is basic .Theorem 9.1 .3 . Let Y and YZ be integrable r .v .'s and Z E then we have(8) c,`(YZ I ) = Zt(Y I -.6-7 ) a .e .[Here "a .e ." is necessary, since we have not stipulated to regard Z as anequivalence class of r .v .'s, although conditional expectations are so regardedby definition . Nevertheless we shall sometimes omit such obvious "a .e .'s"from now on .]