59.10. CONDITIONAL EXPECTATION 1887

Let M consist of all Borel sets of (Rp)n such that∫Rp· · ·∫Rp

XE (x1, · · · ,xn)dλ X1 · · ·dλ Xn =∫(Rp)n

XEdλ (X1,··· ,Xn).

From what was just shown and the definition of (ν1×·· ·×νn) that M contains all setsof the form ∏

ni=1 Ei where each Ei ∈ Borel sets of Rp. Therefore, M contains the algebra

of all finite disjoint unions of such sets. It is also clear that M is a monotone class andso by the theorem on monotone classes, M equals the Borel sets. You could also notethat M is closed with respect to complements and countable disjoint unions and applyLemma 12.12.3. Therefore, the given random vectors are independent and this proves theproposition.

The following Lemma was proved earlier in a different way.

Lemma 59.9.6 If {Xi}ni=1 are independent random variables having values in R,

E

(n

∏i=1

Xi

)=

n

∏i=1

E (Xi).

Proof: By Lemma 59.9.4 and denoting by P the product, ∏ni=1 Xi,

E

(n

∏i=1

Xi

)=

∫R

zdλ P (z) =∫Rn

n

∏i=1

xidλ (X1,··· ,Xn)

=∫R· · ·∫R

n

∏i=1

xidλ X1 · · ·dλ Xn =n

∏i=1

E (Xi).

59.10 Conditional ExpectationDefinition 59.10.1 Let X and Y be random vectors having values in Fp1 and Fp2 respec-tively. Then if ∫

|x|dλ X|y (x)< ∞,

we define

E (X|y)≡∫

xdλ X|y (x).

Proposition 59.10.2 Suppose∫Fp1×Fp2 |x|dλ (X,Y) (x)< ∞. Then E (X|y) exists for λ Y a.e.

y and ∫Fp2

E (X|y)dλ Y =∫Fp1

xdλ X (x) = E (X).

Proof: ∞ >∫Fp1×Fp2 |x|dλ (X,Y) =

∫Fp2

∫Fp1 |x|dλ X|y (x)dλ Y (y) and so∫

Fp1|x|dλ X|y (x)< ∞

, λ Ya.e. Now ∫Fp2

E (X|y)dλ Y