Foundations of the theory of probability

§

2. BorelFieldsofProbability 17

ExtensionTheorem: Itisalways

possible

to

extend

a

non-

negative completely additive set function

P(A),

defined

in

%,

to all sets

of

B%without losing either

of

its

properties (non-

negativeness andcompleteadditivity) and this canbe done

in

onlyoneway.

The extended field B% forms with the extended set func-

tion

P(A)

a

fieldofprobability (B%,P).Thisfieldofprobability

(B%,P)we

shallcalltheBorel

extensionof

the

field($,

P).

The proof of this

theorem, which belongsto the theory of

additive set functions and which sometimes appears in other

forms,canbegivenasfollows:

LetAbeanysubsetofE

;

weshalldenotebyP*(A) the

lower

limitofthesums

y:p(A

n)

n

forall coverings

Acz(SA

n

n

ofthesetAbyafiniteorcountablenumberofsetsA„of$•It

is

easy to prove that P*(A) is then an outer measure in the

Caratheodorysense

2

. Inaccordance withtheCoveringTheorem

(51),

P*(A)

coincides

with

P(A) for

all

setsof

8f.

Itcanbefur-

ther

shownthatallsetsof

$

aremeasurableinthe

Caratheodory

sense.SinceallmeasurablesetsformaBorelfield,allsetsofB%

areconsequently measurable.Thesetfunction P*(A) is,there-

fore,completelyadditiveonB%,and

on

B%wemayset

P(A)

=

P*(A).

Wehavethusshowntheexistenceoftheextension.Theunique-

ness of this extension follows immediately from the minimal

propertyofthefieldB%.

Remark:Even

ifthesets (events) Aof

5

canbeinterpreted

as

actual and (perhapsonlyapproximately) observableevents,

it

doesnot,ofcourse,followfromthisthatthesetsoftheextended

field

B%reasonablyadmitofsuchaninterpretation.

Thusthereisthepossibilitythatwhileafield ofprobability

(5,

P) may be regardedastheimage (idealized, however) of

2

Caratheodory, VorlesungeniiberreelleFunktionen,

pp.

237-258. (New-

York,ChelseaPublishingCompany).