736 CHAPTER 27. ANALYTICAL CONSIDERATIONS

Definition 27.1.5 For µ a probability measure on the Borel sets of Rn,

φ µ (t)≡∫Rn

eit·xdµ.

Theorem 27.1.6 Let µ and ν be probability measures on the Borel sets of Rp andsuppose φ µ (t) = φ ν (t) . Then µ = ν .

Proof: The proof is identical to the above. Just replace λX with µ and λY with ν . ■

27.2 Conditional ProbabilityHere I will consider the concept of conditional probability depending on the theory ofdifferentiation of general Radon measures. This leads to a different way of thinking aboutindependence.

If X,Y are random vectors defined on a probability space having values in Rp1 andRp2 respectively, and if E is a Borel set in the appropriate space, then (X,Y ) is a randomvector with values in Rp1 ×Rp2 and λ (X,Y ) (E×Rp2) = λX (E), λ (X,Y ) (Rp1 ×E) =λY (E). Thus, by Theorem 19.8.1 on Page 520, there exist probability measures, denotedhere by λX|y and λY |x, such that whenever E is a Borel set in Rp1 ×Rp2 ,∫

Rp1×Rp2XEdλ (X,Y ) =

∫Rp1

∫Rp2

XEdλY |xdλX ,

and ∫Rp1×Rp2

XEdλ (X,Y ) =∫Rp2

∫Rp1

XEdλX|ydλY .

Definition 27.2.1 Let X and Y be two random vectors defined on a probabilityspace. The conditional probability measure of Y given X is the measure λY |x in theabove. Similarly the conditional probability measure ofX given Y is the measure λX|y .

More generally, one can use the theory of slicing measures to consider any finite listof random vectors, {X i}, defined on a probability space with X i ∈ Rpi , and write thefollowing for E a Borel set in ∏

ni=1Rpi .∫

Rp1×···×RpnXEdλ (X1,···,Xn)

=∫Rp1×···×Rpn−1

∫Rpn

XEdλXn|(x1,··· ,xn−1)dλ (X1,···,Xn−1)

=∫Rp1×···×Rpn−2

∫Rpn−1

∫Rpn

XEdλXn|(x1,··· ,xn−1)dλXn−1|(x1,··· ,xn−2)dλ (X1,···,Xn−2)

...∫Rp1· · ·∫Rpn

XEdλXn|(x1,··· ,xn−1)dλXn−1|(x1,··· ,xn−2) · · ·dλX2|x1dλX1 . (27.1)

Obviously, this could have been done in any order in the iterated integrals by simply modi-fying the “given” variables, those occurring after the symbol |, to be those which have beenintegrated in an outer level of the iterated integral. For simplicity, write

λXn|(x1,··· ,xn−1) = λXn|x1,··· ,xn−1