Foundations of the theory of probability

70 Appendix

ablesx

n

.IfeventBbelongsto

®,

then,accordingtothe

conditions

ofthetheorem,

P

M

(A)=P(A).

(2)

In the case P(A)

=

our theorem

is

already

true. Let now

P(A) >0.

Then

from (2) followstheformula

Pa(B)

=

P

'{

££I

B)

=P(B),

(3)

and

thereforeP(B) and

P

A

(B)

are

twocompletely

additiveset

functions,

coinciding

on

®

;thereforetheymustremainequalto

eachotheroneverysetoftheBorelextensionB®ofthefieldSt

Therefore, in particular,

P(A)=P

A

(A\=i,

whichprovesourtheorem.

Severalothercases inwhichwecanstatethatcertainprob-

abilitiescanassumeonlythevaluesoneandzero,werediscovered

byP.Levy.
See

P.Lriw,SuruntheoremedeM.Khintchine,Bull,

desSci.Math.v.55,1931,pp.

145-160,TheoremII.