Foundations of the theory of probability

58 VI.Independence;TheLawofLarge

Numbers

(2).

Conversely

sinceP

v

(uczA)

isuniquelydeterminedby

(4)

towithinprobability

zero,thenequation

(2)

followsfrom

(4)

almostcertainly.

Definition 2 :LetMbeasetoffunctions

u^

(I)

of

t

These

functionsarecalledmutuallyindependentintheirtotality

ifthe

following condition is satisfied. Let W and M" be

two non-

intersecting subsetsofM, andletA' (orA")

beasetfrom

g

definedby

a

relationamongu fromM'

(orM");thenwehave

P(A'A")=

P(A')P\A").

TheaggregateofalP«

/t

ofW (orofM") canberegardedas

coordinates of somefunctionv! (or u"). Definition 2 requires

onlytheindependenceofu'andu"inthesenseofDefinition 1 for

eachchoiceofnon-intersectingsetsWandM"

.

Ifu

lt

Mz,...,w

n

aremutuallyindependent,theninallcases

P{u

l

aA

l

,

u

2

cA

2

,...,u

n

czA

n

}

(K)

=

P(«!c

4J

P(«

t

c^

2

).,P(m

b

c^

providedthesets A

A:

belongtothecorresponding

%

{Uk)

(proved

by induction).

This equationis not ingeneral,

however,atall

sufficientforthemutualindependenceofu

lt

u

2

,...,u

n

.

Equation

(5)

iseasilygeneralizedforthecaseofacountably

infiniteproduct.

From themutual independence of u^ ineach finite group

(

w

mi»

u

/*,>

•->u

t*k)

ft doesn°t

necessarily follow that

all u

fl

are

mutuallyindependent.

Finally,itiseasytonotethatthemutualindependenceofthe

functions

u^

isinrealityapropertyofthecorrespondingparti-

tionsty

Ufl

.Further,if u^ aresingle-valuedfunctionsofthecor-

respondingu

fi

,

thenfromthemutualindependenceof

u^

follows

thatofu'.

§

2. IndependentRandomVariables

Ifx

u

x

2 ,

. ..
,

x

n

aremutuallyindependentrandomvariables

thenfromequation
(2)

oftheforegoingparagraphfollows, in

particular,theformula

F^

* *»>

(a

v

a

2

,

..

.

,a

n

)

=

F<**>(a

x

)

F™(a

2

)

.

..F^)(a

n

)

.

(

1

)

//

inthiscasethefield

g

(x

»**••>**)

consistsonly

of

Borelsets

of