Foundations of the theory of probability

2
I.

ElementaryTheory

of

Probability

the system of axioms and in the further
development of the

theory,then thepostulationalconcepts of
a

randomevent and

itsprobabilityseemthemost

suitable.

Thereare

otherpostula-

tionalsystemsofthetheoryof

probability,particularly

those

in

whichtheconceptofprobabilityisnottreatedasoneofthebasic

concepts, but is itself expressed by means of other concepts.

1

However, inthatcase,theaimis different,namely,totieupas

closelyas possiblethemathematical theorywith theempirical

development ofthetheoryofprobability.

§

1.

Axioms

2

LetEbeacollectionofelements

(

t

rj,

£,

..

.

,whichweshallcall

elementaryevents,and

g

asetofsubsetsofE;theelementsof

theset

g

willbecalledrandomevents.

I.

5 isafield

3

of

sets.

II.

g

containsthesetE.

III. ToeachsetAin%isassignedanon-negativerealnumber

P(A).ThisnumberP(A) iscalledtheprobability

of

theeventA.

IV.

P(E) equals 1.

V.

//

AandBhavenoelementin

common,then

P(A

+B)=P(A)+P(B)

Asystemof

sets,

$,

togetherwith a definiteassignmentof

numbersP(A),satisfying

AxiomsI-V,iscalleda

fieldof

prob-

ability.

OursystemofAxiomsI-Visconsistent.This

isprovedbythe

followingexample.LetEconsist

ofthesingleelement$andlet

g

consistofEandthe

nullset0. P(E) isthensetequalto 1 and

P(0) equals0.

1

Forexample,R.von

Mises[l]and

[2]

andS.Bernstein[1].

2

Thereaderwhowishesfromtheoutsettogiveaconcretemeaningto

the

followingaxioms,isreferredto

§

2.

3

Cf.Hausdorff,Mengenlehre,

1927,p.

78.Asystemofsetsiscalledafield

ifthesum,product,anddifferenceoftwosetsofthesystemalsobelongtothe

samesystem.Everynon-emptyfieldcontainsthenullset0.UsingHausdorff's

notation,

we

designatetheproductofAandBbyAB;the

sum

byA

+

B

in

thecasewhereAB

—

0;andinthegeneralcasebyA+B;the

difference

of

AandBbyA-B.ThesetE-A,whichisthecomplementofA,willbedenoted

byK.Weshallassumethatthereaderisfamiliarwiththefundamentalrules

ofoperationsofsetsandtheirsums,products,anddifferences. Allsubsets

of

g

willbedesignatedbyLatincapitals.