Harmonic Analysis on Totally Disconnected Sets
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The
Wiener
Process
A.I.
Probability
Spaces
A probability 2
is
a set,
~
is
negative
countably
property
that
whenever
Ae~
p(B)
=
0
s~ace a
is
p(2)
= i
, p(A)
set
and
of
subsets
that
p
we
space
of
2
: (~ + ~
a meas'ure
B~A
Variables
measure
function
; recall
= O,
of R a n d o m
a complete
~ - algebra
additive
element
A to be
or
Expectat$on
can
(2,
a,
, and
p
, with
the
space
conclude
is
p), is
where
a non-
further
complete
that
B~
if and
.
An to
and
B to
~-A
be
AEa
, the
A~JB
is
an
event
e v e n t ......A
We'll
and we
and
need
the
define
B to be following
the
event
A~B
, and
easy
result
the on
contrary event
A
several
oc-
casions.
Proposition
A.I
Let
{A.} ~ _ ~ J
lira A. ~ ~ j
a.
b,
be
a sequence
vt ~ A = {mgO n n>m
m
lim
A.--- ~ ~ f | A = j ~.J, , n m n>m
bet
of
of
distinct
events.
: m
is
in
infinitely
: ~o
is
in
all
but
all
m
wE
~ n>m
A
{~oc~
Then
many
A,}. j
a finite
num-
A.}
J Proof.
a.
If
~s~
~ n>m
m Thus
for
m=l
is
there
such
that
assume
then
~EA
such
, etc.
there
for
n
nI
n
Conversely
A
2 is
{A
that Hence
n
~eAn
; for m=nl+l i is in i n f i n i t e l y
there
is
w
many
A. J
:j=l,...}
such
that
~eA
n.
b. with
Let the
~
for
be
a fixed
in
all
property
Therefore
~E~ n>m --
Conversely,
that
let
A
m,
but
~ n>m
a finite
~eAn and
n
m~
so
each
j.
n.
J Consequently
for
n2
for
An
, and
number all
~c~ m
of
therefore
A. j
; hence
O ~s~
m
U m n>m
i
An
exists
n> Im h n>m
A
n
1
~eU m
~ n>m
An
; then
there
is
mI
such
that
~s ~ n>m
-- I
An.
162
This
means
that
~aA
whenever
n
n>m -- i q.e.d.
In surely
a probability
(a.s.)
theoretic measure
with
respect
expression p
"
If
space to
"almost S
is
we the
frequently
use
2robability
p
everywhere
a set
then
the
instead
(a.e.)
a random
expression
with
object
almost
of
the
respect
to
X
in
S
measure the is
a func-
tion
(A.1)
Now
X
for
a given
random
object
B
Since we
a
define
is
a
the
o -
(A.I)
~
{B~S
algebra
real-valued
: ~ ÷
it set
is of
~x(B)
- p(x-l(B))
a probability
space
is
clear
that
~
is
a
o -
algebra,
and
function
for
each
and
we
If
B~
the
: B ~
Be8
refer
event
~
. It to
is
this
X-I(B)
{X~B}
S
~
tionally,
(0,i)
the
and
Aca
event
A
A
Generally in
define
immediate space
as
that the
(S,
8,
~X )
distribution
X
Notation
If
we
.
: X-I(B)~a}
WX
as
S
terms
for of
any X
S
, then
~
then is
=
and we
{~
XA
:
~
by
is
a
random
} -
{X A
=
i}
random
object
=
: R(X(~))
X
, if
write
{R(X)}
{~e2
; that
is,
.
S
÷
{X~B}
object
and,
nota-
as
: XA(~)=i
any
denoted
: X(~)eB}
written
{m
is
}
•
R(X)
is
a
relation
163
Now, then
X
is
algebra the
S
is
a random
E
a topological
variable
that
algebra is
(resp., random
S
(recall
Borel
When
if
is
; note
that
is m e a s u r a b l e
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