Foundations of the theory of probability

16 II. InfiniteProbabilityFields

anyfield g.*Wecan, therefore, definetheconcept
ofafield of

probabilityinthefollowingway:LetEbeanarbitraryset,
% a

field
of

subsets

of

E,containingE,and?(A)

anon-negativecom-

pletely
additive

set

functiondefined ongf; thefield

5

together

withtheset

function

?(A)

formsafield

of

probability.

ACoveringTheorem://A,A

lt

A

2 ,

..

.

,A

n

,

...belongto

g

and

Aa(BA

n i (8)

n

then

Proof:

A

=

A<S(A

H

)

=AA

t

+

A

(A

2

• A

2

A

X

)

+

A(A

3

• A

3

A

2

• A

3

AJ+

• 
  
  
  
  • 
    
  
  
  
  

,

n

?{A)

=?(AA

X

)

+

P{A(A

2

• A

2

A,)}

+

...

^

P(^)

+

P(^)

+

••••

§

2. BorelFieldsofProbability

Thefield

5

iscalled aBorel

field,

ifallcountable

sums2^»

ofthesetsA

n

from

gf

belongto

g.

Borel

fieldsare

also

called

com-

pletely

additivesystemsofsets.Fromtheformula

<SA

n

=A

1

+

(A

2

• A

2

A

X)

+

(A

3

• A

3

A

2

• A

Z

A

X

)

+

• •
  (1)

n

wecandeducethat

a

Borelfieldcontains

alsoallthesums <5A

n

n

composedof

a

countablenumber

of

sets

A»

belonging

toit.

From

theformula

%A

n

=E-(BA

n

(2)

n n

thesamecanbesaidfortheproductofsets.

A

field of probability is a Borelfield of probability

if

the

corresponding field

%

isaBorelfield.OnlyinthecaseofBorel

fieldsofprobabilitydoweobtainfullfreedomofaction,without

dangerof theoccurrence of events having no probability. We

shallnowprovethatwemay

limitourselves

to

theinvestigation

ofBorelfieldsofprobability.This

will

followfrom

theso-called

extensiontheorem, towhichweshallnowturn.

Given

a

fieldofprobability

(5,

P).Asisknown

1

,

thereexists

asmallestBorelfieldB^containing5-

And

wehavethe

*

See,forexample,O.Nikodym,Surunegeneralisation

des

integratesde

M.J.Radon,Fund.Math.v.15,1930,p.136.

1

Hausdorff,Mengenlehre,

1927,

p.85.