Foundations of the theory of probability

ChapterI

ELEMENTARY THEORY OF PROBABILITY

Wedefine as elementary theory ofprobability that partof

thetheoryinwhichwehavetodeal withprobabilitiesofonlya

finitenumberofevents.Thetheoremswhichwederiveherecan

beappliedalsototheproblemsconnected

with

an

infinitenumber

of randomevents.However, whenthelatterare

studied, essen-

tiallynewprinciplesareused.Thereforetheonlyaxiomofthe

mathematicaltheoryofprobabilitywhichdealsparticularly

with

thecaseofaninfinitenumberofrandomeventsisnotintroduced

untilthebeginningofChapterII (Axiom VI).

Thetheory ofprobability,

as a

mathematicaldiscipline,can

andshould bedeveloped fromaxioms in

exactlythesame way

asGeometryandAlgebra.Thismeansthatafterwehave

defined

the elements tobe studied andtheir basicrelations, andhave

stated theaxiomsbywhichthese relationsaretobegoverned,

allfurtherexposition
mustbebased

exclusivelyontheseaxioms,

independentoftheusualconcretemeaningofthese

elementsand

theirrelations.

Inaccordancewiththeabove,in

§

1 theconceptofa

fieldof

probabilitiesisdefinedasasystemofsetswhichsatisfiescertain

conditions.Whattheelementsofthissetrepresentis ofnoim-

portanceinthe

purelymathematical developmentofthe

theory

ofprobability

(cf.theintroduction ofbasicgeometric concepts

intheFoundations

of

GeometrybyHilbert,orthedefinitionsof

groups,ringsandfieldsinabstractalgebra).

Everyaxiomatic (abstract) theoryadmits,

as

iswellknown,

ofanunlimitednumberofconcreteinterpretationsbesidesthose

fromwhichit

wasderived.Thuswefindapplicationsinfields

of

science which

havenorelationtotheconceptsofrandomevent

andof

probabilityintheprecisemeaningofthesewords.

Thepostulationalbasis of the theory of probability can be

established bydifferent methods in respect totheselection of

axiomsaswellasintheselectionofbasicconcepts

and

relations.

However,ifouraimistoachievetheutmostsimplicity

bothin