1886 CHAPTER 59. BASIC PROBABILITY

Proof: First let E be a Borel set in Rk. From the definition,

λ g(X1,··· ,Xn) (E) = P(g(X1, · · · ,Xn) ∈ E)

= P((X1, · · · ,Xn) ∈ g−1 (E)

)= λ (X1,··· ,Xn)

(g−1 (E)

)∫Rk

XEdλ g(X1,··· ,Xn) =∫Rp1×···×Rpn

Xg−1(E)dλ (X1,··· ,Xn)

=∫Rp1×···×Rpn

XE (g(x1, · · · ,xn))dλ (X1,··· ,Xn).

This proves 59.9.15 in the case when h is XE . To prove it in the general case, approximatethe nonnegative Borel measurable function with simple functions for which the formula istrue, and use the monotone convergence theorem.

It remains to prove the last assertion that functions of independent random vectors arealso independent random vectors. Let E be a Borel set in Rk1 ×·· ·×Rkn . Then for

π i (x1, · · · ,xn)≡ xi,∫Rk1×···×Rkn

XEdλ (g1(X1),··· ,gn(Xn))

≡∫Rp1×···×Rpn

XE ◦ (g1 ◦π1, · · · ,gn ◦πn)dλ (X1,··· ,Xn)

=∫Rp1· · ·∫Rpn

XE ◦ (g1 ◦π1, · · · ,gn ◦πn)dλ Xn · · ·dλ X1

=∫Rk1· · ·∫Rkn

XEdλ gn(Xn) · · ·dλ g1(X1)

and this proves the last assertion.

Proposition 59.9.5 Let ν1, · · · ,νn be Radon probability measures defined on Rp. Thenthere exists a probability space and independent random vectors {X1, · · · ,Xn} defined onthis probability space such that λ Xi = ν i.

Proof: Let (Ω,S ,P) ≡ ((Rp)n ,S1×·· ·×Sn,ν1×·· ·×νn) where this is just theproduct σ algebra and product measure which satisfies the following for measurable rect-angles.

(ν1×·· ·×νn)

(n

∏i=1

Ei

)=

n

∏i=1

ν i (Ei).

Now let Xi (x1, · · · ,xi, · · · ,xn) = xi. Then from the definition, if E is a Borel set in Rp,

λ Xi (E)≡ P{Xi ∈ E}

= (ν1×·· ·×νn)(Rp×·· ·×E×·· ·×Rp) = ν i (E).