Foundations of the theory of probability

30

III.

RandomVariables

Proof. Giventhedistributionfunctions

^

1/

u

t

...

/

.

B

, satisfying

thegeneralconditionsofChap.II,

§

3,IIIandalsoconditions

(2)

and

(3).

Everydistributionfunction

&&&...p.

definesuniquely

acorrespondingprobabilityfunction P^^,...^ forallBorelsets

ofR

n

(cf.

§3).

WeshalldealinthefutureonlywithBorelsets

ofR

n

andwith BorelcylindersetsinE.

Foreverycylinderset

weset

PW=

P*,*,...,^

V

).

(4)

SincethesamecylindersetAcanbedennedbyvarioussetsA',

wemust first show that formula

(4)

yields always the same

value

for

P(A).

Let (x^,x^...,

XpJ

be afinitesystem of randomvariables

Xp.

Proceedingfromtheprobabilityfunction P^^,...^ ofthese

randomvariables, wecan, inaccordancewith therulesin

§3,

define the probability function P^^...^ of each subsystem

(x

Hi

,

x

H

,

.

.

.,x

/H

)

. Fromequations
(2)

and

(3)

itfollowsthat

thisprobabilityfunction definedaccordingto

§

3 isthesameas

thefunctionP^^

2

...

Hlt

given

a

priori.

We

shallnowsupposethat

thecylindersetA isdefinedbymeansof

A=p;l„

it

...

H

y)

and simultaneouslyby means of

where all random variables x

M

and

*

belong to the system

(

x

/*i

>

x

ht

>••

• 
  

»

*«J

»

whichisobviouslynotanessentialrestriction.

The

conditions

and

(V

,

V

, ...,*« )cA"

areequivalent. Therefore

P

^\

H

%

• ••
  H
  k

(

A')

=

P

^«••n*

{(^»

*/4,

• 
  

*'

'

>

X

H

k

)

c^'}

=

P^,...^{(*>V

X'

'•"

**J

cA

l

=

%^'^J

A

^

>

whichproves our statement concerningthe uniqueness of the

definitionof P(A).