Foundations of the theory of probability

Appendix

ZERO-OR-ONE LAW IN THE THEORY

OF PROBABILITY

We have noticed several cases in which certain limiting

probabilitiesarenecessarilyequaltozero orone. Forexample,

theprobabilityofconvergenceofaseriesofindependentrandom

variablesmayassumeonlythesetwovalues

1

.Weshallprovenow

a generaltheorem includingmanysuchcases.

Theorem:Let

x

u

x

z,

..

.

,x

n

,

..

.

beanyrandomvariables

and

let

f(Xi,

x

2

,... ,x

n

,.. .) bea

Baire

function

2

ofthe variables

x

Xt

x

2 ,

...

,

x„,...suchthattheconditionalprobability

P*.*.....*{/(*)=

0}

of

therelation

f{x

1

,x

2>

...,x

n

,...)

=0

remains,

whenthefirstnvariablesx

lf

x

2 ,

..

.

,x„are

known,

equal

to theabsolute probability

P{/(*)=0} (1)

for

every n. Under these conditionsthe probability

(1)

equals

zeroorone.

Inparticular,theassumptionsofthistheoremarefulfilledif

thevariables

x

n

are

mutually

independentand

if

the

valueofthe

function
f(x)

remainsunchangedwhenonlyafinitenumberof

variables arechanged.

Proof

of

theTheorem:LetusdenotebyAtheevent

f(x)

=0.

Weshallalsoinvestigatethefield

St

ofallevents whichcanbe

definedthroughsomerelations among

a

finite numberofvari-

1

Cf.Chap.VI,§5.Thesamethingistrueoftheprobability

PK-rf„-*o}

inthestronglawoflargenumbers;atleast,whenthevariablesx

n

aremutu-

allyindependent.

2

ABairefunctionisonewhichcanbeobtainedbysuccessivepassagesto

thelimit,ofsequencesoffunctions,startingwithpolynomials.

69