Foundations of the theory of probability

Chapter

II

INFINITE PROBABILITY FIELDS

§

1. AxiomofContinuity

Wedenote

by 2)A

m

,asis

customary,

theproductofthe

sets

m

A
m

(whetherfiniteorinfiniteinnumber)andtheirsumby <5A

m

.

m

OnlyinthecaseofdisjointsetsA

m

istheform

^A

m

used

instead

m

of <&A
m

. Consequently,

m

®A

m

=A

1

+

A

t

+

•;

ZAm

=A

1

+

A

2

+---,

m

^A

m

=A

1

A

2

"-.

Inallfutureinvestigations,weshallassumethatbesidesAxioms

I

• V,still anotherholdstrue

:

VI. For

adecreasing sequenceofevents

A

1

z)A

2

^-"

3^

n

z>.-. (1)

of

&

forwhich

®A

»

=

,

(2)

the

following

equation holds:

limP

(4n)

=

. w-*oo

(3)

Inthefutureweshalldesignate byprobabilityfieldonlya

field ofprobabilityas outlined inthefirst chapter, whichalso

satisfiesAxiomVI.Thefieldsofprobability

as

definedinthefirst

chapterwithoutAxiom

VImightbecalled

generalized

fieldsof

probability.

Ifthesystem

J

ofsetsisfinite,AxiomVIfollowsfromAxioms

I

• V.Foractually,inthatcasethereexistonlyafinitenumber

of different sets in the sequence

(1).

Let A

k

be the smallest

amongthem,thenallsetsA^coincidewithA

k

andweobtainthen

14