Foundations of the theory of probability

§1. AxiomofContinuity 15

n

limP(^

B)

=P(o)=

0.

Allexamplesof

finite

fieldsofprobability, inthefirstchapter,

satisfy,therefore, AxiomVI.The systemofAxioms I

• VIthen

provestobeconsistentandincomplete.

Forinfinitefields,ontheotherhand,theAxiomofContinuity,

VI,provedtobeindependentofAxiomsI

• V.Sincethenewaxiom

isessentialforinfinitefieldsofprobabilityonly,itisalmostim-

possibletoelucidateitsempiricalmeaning,
as

hasbeendone,for

example,inthecaseofAxioms I

• 
  

V

in

§

2 ofthefirstchapter.

For,indescribinganyobservablerandomprocesswe

can

obtain

onlyfinitefieldsofprobability.Infinitefieldsofprobabilityoccur

onlyasidealizedmodels

of

real

randomprocesses.Welimitour-

selves,arbitrarily,to
onlythosemodelswhichsatisfyAxiomVI.

This limitation hasbeenfound expedient in researches of the

mostdiversesort.

GeneralizedAdditionTheorem:

//

A

lt

A,,...

,

A

n,

.. .and

Abelongto
ft,

then

from

A=Z

A

n

(4)

followstheequation

Proof: Let

Then,obviously
^(R
n

)

=

0,

n

and,therefore,
accordingtoAxiomVI

limP(R

n)

= fi-»oo

• (6)

Onthe
otherhand,bytheadditiontheorem

P(A)

=

P(A

1 ) +

P(A

2

) +...+P(A

n) +

P(R

n)

. (7)

From

(6)

and

(7) weimmediatelyobtain

(5).

Wehaveshown,

then, that theprobability

P(A) isa com-

pletelyadditive
setfunctionon
$.

Conversely,

AxiomsVandVI

holdtrue
for everycompletelyadditive
setfunctiondefinedon

n

P{A)=2P(A

n

).

(5)

n