Foundations of the theory of probability

32

III.RandomVariables

occur,i.e.

^

=

^,.

..,.»(£»)•

Forbrevityweset

^,

t...Mn(B)

=

P

n

(B);

then, obviously

P

n

(B

n)

=?(A

n

)

^L>0.

IneachsetB

n

itispossibletofindaclosedboundedsetU

n

such

that

P»(B

n

-U

n)^-^.

Fromthisinequalitywehave

for

theset

theinequality

Let,

morever,

"

r

1*1ft•••f*H

V

"

P(A

n

-V

n

)^J-.

(5)

w

n

=

v

x

v

2

...v

n

.

From

(5)

itfollowsthat

P(A

n

-W

n)

g

€

.

Since W

n

cV

n

c:A

n

,

itfollowsthat

P(W

n

)^P(A

n

)-e^L-8.

Ife is sufficiently small,

P(W

n)

>

and

W

n

is not

empty. We

shallnowchooseineachsetW

n

apoint

£

U)

withthecoordinates

a» Everypoint

^

M

+^),

p

=

0, 1,2,

...

,

belongstothesetV

n

;

therefore

(*r

p)

.

*;r

p)

*<

n

.

+

»)

=

^....,.(f<»^»)

ct/„

.

SincethesetsU

n

areboundedwemay(bythediagonalmethod)

choosefromthesequence {£

(n)

}

asubsequence

for whichthecorrespondingcoordinates
*2?

tendforanyA:to

adefinite limit x

k

. Let, finally,
|

be a point in set£7 with the

coordinates

X

t*k

=

x

k

>

x,*=

0, /*

+

/**• £=1,2,3,...