Foundations of the theory of probability

18 II. InfiniteProbabilityFields

actualrandomevents,the extendedfield

ofprobability (B%, P)

willstill remainmerelyamathematical
structure.

Thus setsof

B%

are

generallymerelyideal eventstowhich

nothingcorrespondsin
theoutsideworld.However,ifreasoning

which
utilizestheprobabilitiesofsuchidealeventsleadsustoa

determinationofthe probabilityof anactual eventof

g,

then,

from an empirical point of view also, this determination will

automaticallyfailtobecontradictory.

§

3. ExamplesofInfiniteFieldsofProbability

I. In

§

1 of thefirst chapter, wehave constructed various

finiteprobabilityfields.

Letnow

E

=

{£x

,

£ 2

>•••>

ln»

• •}

bea

countable

set,andlet

5

coincidewiththeaggregateofthesubsetsofE.

Allpossibleprobability fieldswithsuchanaggregate

5

are

obtainedinthefollowingmanner:

We

take

a

sequenceofnon-negativenumbers

p„,

suchthat

Pi+Vi+

...

+

Vn

+•••

= 1

andforeachsetAput

P(A) -

2'fin,

n

wherethesummation

2'

extendstoalltheindicesn for

which

$n

belongs toA. Thesefields ofprobabilityareobviously Borel

fields.

II. In thisexample, weshallassumethat E represents the

realnumber
axis. Atfirst,let
g

beformedof

allpossiblefinite

sums of half-open intervals [a; b)

— {a£.tj<b}

(taking into

considerationnotonlytheproperintervals,withfiniteaandb,

butalsotheimproper intervals [-

<x>

;

a), [a,-+ oo) and [-o©j

4-oo

)

).

g

isthenafield.Bymeansoftheextensiontheorem,how-

ever, eachfieldofprobabilityon
5

canbeextendedtoasimilar

field onB%. The system
of sets

B% is, therefore, in our case

nothingbutthesystemofall

Borelpointsets on

a

line. Letus

turnnowtothefollowingcase.

III. AgainsupposeEtobetherealnumber

axis,while

g

is

composedofallBorelpointsetsofthisline.Inordertoconstruct

a field of probability with thegiven field
gf,

it is sufficientto

defineanarbitrarynon-negativecompletelyadditiveset-function