Foundations of the theory of probability

28 III. RandomVariables

/(**.**.-••»**,)=-<).

(1)

Inordertodetermine

an

arbitrary

cylinderset P

Ml

^

...

^

(A')

by

sucharelation,weneedonlytakeas

/

afunctionwhichequals

onA',butoutsideofA'equalsunity.

AcylindersetisaBorelcylindersetifthecorrespondingset

A

f

isaBorelset.AllBorelcylindersets

of

thespaceR

M

forma

field,

whichweshallhenceforthdenote

by

g

M4

.

TheBorel

extensionofthefield

%

M

weshalldenote,asalways,

byB%

M

.SetsinB%

M

weshallcallBorelsets

of

thespaceR

M

.

Lateronweshallgiveamethodofconstructingandoperating

withprobabilityfunctionson

%

M

,

andconsequently,bymeansof

theExtensionTheorem,onB%

M

also.Weobtaininthismanner

fieldsofprobabilitysufficientforallpurposesinthecasethatthe

set M is denumerable. We can therefore

handle all

questions

touchinguponadenumerablesequenceofrandomvariables.

But

ifMisnotdenumerable,manysimpleandinterestingsubsetsof

R

M

remainoutsideofB%

M

.Forexample,thesetofallelements

£

for which *

M

remains smaller than a fixed constant for all

indices

/*,

does not belong tothe system B%

M

if the set M is

non-denumerable.

Itisthereforedesirableto

trywheneverpossibletoputeach

probleminsuchaformthatthespaceofallelementary

events

£

hasonlyadenumerablesetof coordinates.

Letaprobabilityfunction P(A) be definedon

%

M

. Wemay

then regard every coordinate %

M

of the elementary event

£

as a

random variable. In consequence, every finite group

(

x

rii>

x

m»>

• ••*x

fJ

°f these

coordinates has an ^-dimensional

probabilityfunction P^....^^) anda

correspondingdistribu-

4

FromtheaboveitfollowsthatBorelcylindersets

areBorelsetsdefinable

byrelationsoftype

( 1

).NowletAandBbetwoBorelcylindersetsdefined

bytherelations

/(*/*i.*t*t *#«J

=

0» Sfai.*l

X

U)

=

• 
  

Thenwe

can

definethesetsA+B,AB,andA-Brespectivelybytherelations

f-g

=0,

f*+g

2

=

0,

where

a>(x)= for x4= and

w

(0)

= 1 If

/

and

g

areBorelfunctions,

so

also

are

f-g,f

+

g

2

and

f

+ <o{g)

;

therefore,A+B,ABandA-BareBorel

cylindersets.Thuswe

haveshownthatthesystemofsets $

3f

isafield.