Kenneth Kuttler

59.9. CONDITIONAL PROBABILITY 1885

Thus, using this in the above,∫Rp1

∫Rp2

∫Rp3

XE (x)XF (y)XG (z)dλ Zdλ Ydλ X

=∫Rp1

∫Rp2

∫Rp3

XE (x)XF (y)XG (z)dλ Z|xydλ Ydλ X

and also it reduces to ∫Rp1×Rp2

∫Rp3

XE (x)XF (y)XG (z)dλ Zdλ (X,Y)

=∫Rp1×Rp2

∫Rp3

XE (x)XF (y)XG (z)dλ Z|xydλ (X,Y)

Now by uniqueness of the slicing measures again, for λ (X,Y) a.e. (x,y) , it follows that

λ Z = λ Z|xy

Similar conclusions hold for λ X,λ Y. In each case, off a set of measure zero the distributionmeasures equal the slicing measures.

Conversely, if the distribution measures equal the slicing measures off sets of measurezero as described above, then it is obvious that the random variables are independent. Thesame reasoning applies for any number of random variables.

Thus this gives a different and more analytical way to think of independence of finitelymany random variables. Clearly, the argument given above will apply to any finite set ofrandom variables.

Proposition 59.9.3 Equations 59.9.14 and 59.9.13 hold with XE replaced by any non-negative Borel measurable function and for any bounded continuous function or for anyfunction in L1.

Proof: The two equations hold for simple functions in place of XE and so an appli-cation of the monotone convergence theorem applied to an increasing sequence of simplefunctions converging pointwise to a given nonnegative Borel measurable function yields theconclusion of the proposition in the case of the nonnegative Borel function. For a boundedcontinuous function or one in L1, one can apply the result just established to the positiveand negative parts of the real and imaginary parts of the function.

Lemma 59.9.4 Let X1, · · · ,Xn be random vectors with values in Rp1 , · · · ,Rpn respectivelyand let g : Rp1 × ·· ·×Rpn → Rk be Borel measurable. Then g(X1, · · · ,Xn) is a randomvector with values in Rk and if h : Rk→ [0,∞), then∫

Rkh(y)dλ g(X1,··· ,Xn) (y) =∫

Rp1×···×Rpnh(g(x1, · · · ,xn))dλ (X1,··· ,Xn). (59.9.15)

If Xi is a random vector with values in Rpi , i = 1,2, · · · and if gi : Rpi → Rki , where gi isBorel measurable, then the random vectors gi (Xi) are also independent whenever the Xiare independent.

59.9. CONDITIONAL PROBABILITY 1885Thus, using this in the above,Lo bo. [ne 2@8) Ze) Me (a) ddzddydaxJR?P1 JR?2 JR?3= [ [ [ ALE (x) An (y) 4G (z) dAzxydAydaxJR?1 JIRP2 JRP3and also it reduces to| Xp (x) Xe (y) Xo (2) daadA xy)R?1xR?2 JR?3~ L. xRP2 bon Xe (x) Re (y) 2G (2) dAzixyda xy)Now by uniqueness of the slicing measures again, for A (x y) a.e. (x,y), it follows thatAq= NalxySimilar conclusions hold for Ax, Ay. In each case, off a set of measure zero the distributionmeasures equal the slicing measures.Conversely, if the distribution measures equal the slicing measures off sets of measurezero as described above, then it is obvious that the random variables are independent. Thesame reasoning applies for any number of random variables.Thus this gives a different and more analytical way to think of independence of finitelymany random variables. Clearly, the argument given above will apply to any finite set ofrandom variables.Proposition 59.9.3. Equations 59.9.14 and 59.9.13 hold with 2_ replaced by any non-negative Borel measurable function and for any bounded continuous function or for anyfunction in L!.Proof: The two equations hold for simple functions in place of 2 and so an appli-cation of the monotone convergence theorem applied to an increasing sequence of simplefunctions converging pointwise to a given nonnegative Borel measurable function yields theconclusion of the proposition in the case of the nonnegative Borel function. For a boundedcontinuous function or one in L', one can apply the result just established to the positiveand negative parts of the real and imaginary parts of the function.Lemma 59.9.4 Let X,,---,X;, be random vectors with values in R?!,--+ ,IR?" respectivelyand let g: RP! x »-. x R?» — R* be Borel measurable. Then g(X,--+ ,Xn) is a randomvector with values in R* and if h: Rk — [0,), thenek h(y) dAgx,,....x,) VY) =/ h(g(%142++ +Xn)) OA (x o-X,): (59.9.15)R?1 x-++xRPnIf X; is a random vector with values in R?,i = 1,2,--- and if g; : R?' + Ri, where g; isBorel measurable, then the random vectors g;(X;) are also independent whenever the X;are independent.