2.∫XdP=∫ Zdp for each G∈g Using this result, we define E[XIg=Z Proof. The Radon-Nikodym theorem Remark. Since the conditional expectation is only as.(P) unique, most of the equations below strictly speaking need a qualifying a.s. (P)' appended to them to be true. But since this is a bit tedious, we adopt instead the convention that the statement X= Y means P(wEQ: X(w)=Y(w)))=1. If two random variables W and Z both qualify as the conditional expectation E[XIg], then we will sometimes call them versions of E[Xg This Cl-based definition can be intuitively motivated independently of the projection- based definition in the following way. On events such that we know whether they have occurred, our best guess of X should track X perfectly In any case, it had better be true that our two definitions of the conditional expec tation coincide when they both apply, i. e. on C nc2=C2. They do. You may want to try to prove this for yourself Having defined the conditional expectation with respect to a o-algebra, we now define the conditional expectation with respect to a family of stochastic variables Definition. Let Y E L(Q, F, P) and let (Xa, aE I be a family of random vari- ables. Then the conditional expectation E Y Xa, CEIl is defined as E Ylo Xa,aEIl Since E X is a o(X)-measurable random variable, there is a borel function f such that E[YIX=f(X). Sometimes we use the notation f()=EYIX=a2. R G XdP = R G ZdP for each G ∈ G. Using this result, we define E [X|G] = Z. Proof. The Radon-Nikodym theorem. Remark. Since the conditional expectation is only a.s. (P) unique, most of the equations below strictly speaking need a qualifying ‘a.s. (P)’ appended to them to be true. But since this is a bit tedious, we adopt instead the convention that the statement X = Y means P ({ω ∈ Ω : X (ω) = Y (ω)}) = 1. If two random variables W and Z both qualify as the conditional expectation E [X|G] , then we will sometimes call them versions of E [X|G]. This L 1 -based definition can be intuitively motivated independently of the projection￾based definition in the following way. On events such that we know whether they have occurred, our best guess of X should track X perfectly. In any case, it had better be true that our two definitions of the conditional expec￾tation coincide when they both apply, i.e. on L 1 ∩ L2 = L 2 . They do. You may want to try to prove this for yourself. Having defined the conditional expectation with respect to a σ–algebra, we now define the conditional expectation with respect to a family of stochastic variables. Definition. Let Y ∈ L 1 (Ω, F, P) and let {Xα, α ∈ I} be a family of random vari￾ables. Then the conditional expectation E [Y | {Xα, α ∈ I}] is defined as E [Y |σ {Xα, α ∈ I}] Since E [Y |X] is a σ (X) −measurable random variable, there is a Borel function f such that E [Y |X] = f (X). Sometimes we use the notation f (x) = E [Y |X = x] 7