Basic Properties of Measureable Sets
Math 362 – Spring 2002
R. Pruim
[email protected]
Calvin College
Basic Properties of Measureable Sets – p.1/12
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Basic Properties of Measureable Sets – p.2/12
is countably additive on
.
The proof consists of establishing 7 properties:
is a ring.
.
.
, then
and
are disjoint, elementary sets.
is countably additive on
.
, then
if
is additive on
4.
5. If
and each
3.
2. If
1.
Proof.
6.
is a -ring and
Thm.
Thm 11.10
.
7.
is closed under countable unions and difference (so
Basic Properties of Measureable Sets – p.3/12
by showing that
We can show that that
. Then there are elementary sets
such that
and
.
Let
and
is a ring
1.
Basic Properties of Measureable Sets – p.4/12
by showing that
We can show that that
. Then there are elementary sets
such that
and
.
Let
and
is a ring
1.
as
(3)
(3)
reason for (2):
(check using a membership table.)
The proof for differences is similar.
(3)
Note that
Basic Properties of Measureable Sets – p.4/12
2.
)
), so
(by sub-additivity of
and each
"!
#$ (where
Basic Properties of Measureable Sets – p.5/12
2.
) we see that
, so
and
! $#
), so
(by sub-additivity of
Similarly (reversing roles of
)
and each
"!
#$ (where
Basic Properties of Measureable Sets – p.5/12
2.
(by sub-additivity of
) we see that
, so
and
! $#
), so
Similarly (reversing roles of
)
and each
"!
#$ (where
%
as
.
# %
Combining these we get
Basic Properties of Measureable Sets – p.5/12
2.
(by sub-additivity of
) we see that
, so
and
! $#
), so
Similarly (reversing roles of
)
and each
"!
#$ (where
%
.
for all
This implies that
as
Note:
# %
Combining these we get
.
Basic Properties of Measureable Sets – p.5/12
is additive on
be convergent sequences of
&
!
&
! we get:
! Letting
We know:
!
Let
and
elementary sets.
3.
.
Basic Properties of Measureable Sets – p.6/12
is additive on
be convergent sequences of
&
!
%
%
+
if
, if we choose
,
+
+
,
+
,
.
+-
%
&
#$ +
&
$# %
# %
%
# !
%
So
,
# &
# %
%
%
*)('
Recall that we just showed that
and each
. So for any
large enough,
,
.
&
! ! we get:
Letting
We know:
!
Let
and
elementary sets.
3.
Basic Properties of Measureable Sets – p.6/12
is additive on
be convergent sequences of
&
!
&
! we get:
.
!
, so
&
, then
&
If
.
! Letting
We know:
!
Let
and
elementary sets.
3.
.
Basic Properties of Measureable Sets – p.6/12
99
7
9 !
5 ! 342
1 5
1
with
: Compare
99
87
9:9
9
78
9 !
5 !
0
6
5 (if
<;
*)('
0
0
/
4.
are disjoint, elementary sets)
: follows from sub-additivity
, so
, so
.
Basic Properties of Measureable Sets – p.7/12
=
>
where
Let
/
&
5.
. WLOG the
are disjoint.
Basic Properties of Measureable Sets – p.8/12
=
>
9
are disjoint.
78
99 5 If
, then define
get a disjoint union.
# ?
. WLOG the
where
?
Let
/
&
5.
to
Basic Properties of Measureable Sets – p.8/12
=
>
, then
are disjoint.
. WLOG the
78
9:9
9
Let
where
5 Let
/
&
5.
Basic Properties of Measureable Sets – p.8/12
=
>
are disjoint.
is a ring of sets.
3@ <
B5
@
3 @A<
B5
@
since
78
9:9
9
. WLOG the
, then
Let
where
5 Let
/
&
5.
Basic Properties of Measureable Sets – p.8/12
=
>
, so . . .
are disjoint.
78
9:9
9
, then
. WLOG the
...
Let
where
5 Let
/
&
5.
.
Basic Properties of Measureable Sets – p.8/12
=
>
78
9:9
9
, so . . .
are disjoint.
.
Exercise: use this to show that
1
) with
C
(as
there is
So for each
1
, but we only know
We would be done if each
.
...
, then
. WLOG the
Let
where
5 Let
/
&
5.
.
Basic Properties of Measureable Sets – p.8/12
=
>
78
9:9
9
, so . . .
are disjoint.
.
Exercise: use this to show that
1
) with
C
(as
there is
So for each
1
, but we only know
We would be done if each
.
...
, then
. WLOG the
Let
where
5 Let
/
&
5.
.
Basic Properties of Measureable Sets – p.8/12
with
and
Let
is countably additive on
6.
disjoint.
, so
for all
then each
then
2.
for some
1.
Consider 2 cases:
by (4)
Basic Properties of Measureable Sets – p.9/12
We need to show that
is -ring
7.
is:
closed under countable unions
closed under difference
Basic Properties of Measureable Sets – p.10/12
We need to show that
is -ring
7.
is:
HG 1
DE1
5
D HGF
1
5
D1
5
HG 1
closed under countable unions
Let
be a countable collection of sets from
So
where each
.
But then
is a countable union of
sets in
, so
closed under difference
Basic Properties of Measureable Sets – p.10/12
is -ring
7.
We need to show that
is:
closed under countable unions
with
.
, so it suffices to show
,
# # # # Let
closed under difference
.
# & &
# # &
& , so it suffices to show that
each
.
, so it suffices to show that
.
, so we’re done.
Basic Properties of Measureable Sets – p.10/12
Appendectory Notes Follow
Basic Properties of Measureable Sets – p.11/12
Membership Table
1
0
1
0 0
0
1
1
0
1
1
0 0
0
1
1
1
0
1
0 0
0
1
1
0
0
0
1 0
0
1
1
0
0
0
0 1
0
1
1
0
0
0
1 1
0
1
1
Note that in all missing rows of the table, there would be a 0
.
in the column labeled
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Basic Properties of Measureable Sets – p.12/12