Singular Measures
David C. Ullrich
D mu(x) = lim_{r -> 0} mu(B(x,r)) / m(B(x,r))
for all x for which the limit exists; here m is Lebesgue
measure.
Say mu is "singular" if it is singular with respect
to m, ie mu and m are mutually singular (concentrated
on disjoint sets).
It's well known that if mu is singular then
(*) D mu(x) = 0 ae[m].
It also seems to be true that if mu is singular then
(**) D mu(x) = infinity ae[mu].
I mention this because it's something I didn't know
until the other day when I proved it while preparing
for class - seems at least possible that there's someone
else out there who didn't know it either. (It also seems
possible that the proof below is wrong; if anyone sees
anything funny that would be great.) The two books I
know are Folland and Rudin. Folland doesn't mention
the issue at all, while I was suspicious of my proof
of (**) because I recalled that Rudin warned that (*)
is false. But I looked it up - in fact Rudin points
out that a version of (**) with an _assymetric_ sort
of "derivative" in place of D is false.
So (**) doesn't contradict what's in Rudin. But I
remain suspicious, since it seems curious that
neither Folland nor Rudin mentions the true version
of (**) even though the proof is more or less the
same as the proof of (*), mutatis mutandis.
So let me know where the error is:
Proof of (**): Say
omega(x) = liminf_{r -> 0} mu(B(x,r)) / m(B(x,r)).
We need to show that omega = infinity ae[mu].
Fix L in (0, infinity); by countable additivity
it's enough to show that
mu({x : omega(x) < L}) = 0.
Say K is a compact subset of {x in S : omega(x) < L}.
Since mu(T) = 0 and mu is regular it's enough to
show that mu(K) = 0.
Let epsilon > 0. Choose an open set V which
contains S, such that m(V) < epsilon.
Now for every x in K there exists r = r(x) > 0 such
that B(x,r) is contained in V and
mu(B(x,3r)) / m(B(x,3r)) < L.
By the standard covering lemma there exist x_1,...,
x_k in K and r_1, ..., r_k > 0 such that the balls
B(x_j,r_j) are disjoint, B(x_j, r_j) is contained
in V, and K is contained in the union of B(x_j,3r_j).
So
mu(K) <= sum mu(B(x_j,3r_j))
<= L sum m(B(x_j,3r_j))
= 3^n L sum m(B(x_j,r_j))
= 3^n L m(union B(x_j,r_j))
<= 3^n L m(V)
< 3^n L epsilon.
Let epsilon -> 0 and it follows that mu(K) = 0. QED.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Rotwang
I don't think you mentioned S and T before, but I guess they are
supposed to be a partition of R^n with m(S) = mu(T) = 0, right? (If
you did mention them and I've somehow missed it then I apologise, I
have been awake for an awfully long time so it's quite possible.)
> Let epsilon > 0. Choose an open set V which
> contains S, such that m(V) < epsilon.
>
> Now for every x in K there exists r = r(x) > 0 such
> that B(x,r) is contained in V and
>
> mu(B(x,3r)) / m(B(x,3r)) < L.
>
> By the standard covering lemma there exist x_1,...,
> x_k in K and r_1, ..., r_k > 0 such that the balls
> B(x_j,r_j) are disjoint, B(x_j, r_j) is contained
> in V, and K is contained in the union of B(x_j,3r_j).
> So
>
> mu(K) <= sum mu(B(x_j,3r_j))
>
> <= L sum m(B(x_j,3r_j))
>
> = 3^n L sum m(B(x_j,r_j))
>
> = 3^n L m(union B(x_j,r_j))
>
> <= 3^n L m(V)
>
> < 3^n L epsilon.
>
> Let epsilon -> 0 and it follows that mu(K) = 0. QED.
Looks correct to me, not that that ought to count for much.
David C. Ullrich
<sg...@hotmail.co.uk> wrote:
>On 12 Nov, 11:23, David C. Ullrich <dullr...@sprynet.com> wrote:
>> Suppose mu is a regular Borel measure on R^n. Define
>>
>> � D mu(x) = lim_{r -> 0} mu(B(x,r)) / m(B(x,r))
>>
>> for all x for which the limit exists; here m is Lebesgue
>> measure.
>>
>> Say mu is "singular" if it is singular with respect
>> to m, ie mu and m are mutually singular (concentrated
>> on disjoint sets).
>>
>> It's well known that if mu is singular then
>>
>> (*) � D mu(x) = 0 ae[m].
>>
>> It also seems to be true that if mu is singular then
>>
>> (**) �D mu(x) = infinity ae[mu].
>>
>> I mention this because it's something I didn't know
>> until the other day when I proved it while preparing
>> for class - seems at least possible that there's someone
>> else out there who didn't know it either. (It also seems
>> possible that the proof below is wrong; if anyone sees
>> anything funny that would be great.) The two books I
>> know are Folland and Rudin. Folland doesn't mention
>> the issue at all, while I was suspicious of my proof
>> of (**) because I recalled that Rudin warned that (*)
That last (*) was a typo for (**).
>> is false. But I looked it up - in fact Rudin points
>> out that a version of (**) with an _assymetric_ sort
>> of "derivative" in place of D is false.
>>
>> So (**) doesn't contradict what's in Rudin. But I
>> remain suspicious, since it seems curious that
>> neither Folland nor Rudin mentions the true version
>> of (**) even though the proof is more or less the
>> same as the proof of (*), mutatis mutandis.
>>
>> So let me know where the error is:
>>
>> Proof of (**): Say
>>
>> � omega(x) = liminf_{r -> 0} mu(B(x,r)) / m(B(x,r)).
>>
>> We need to show that omega = infinity ae[mu].
>> Fix L in (0, infinity); by countable additivity
>> it's enough to show that
>>
>> � mu({x : omega(x) < L}) = 0.
>>
>> Say K is a compact subset of {x in S : omega(x) < L}.
>> Since mu(T) = 0 and mu is regular it's enough to
>> show that mu(K) = 0.
>
>I don't think you mentioned S and T before, but I guess they are
>supposed to be a partition of R^n with m(S) = mu(T) = 0, right?
Yes, sorry.
>(If
>you did mention them and I've somehow missed it then I apologise, I
>have been awake for an awfully long time so it's quite possible.)
>
>
>> Let epsilon > 0. Choose an open set V which
>> contains S, such that m(V) < epsilon.
>>
>> Now for every x in K there exists r = r(x) > 0 such
>> that B(x,r) is contained in V and
>>
>> � �mu(B(x,3r)) / m(B(x,3r)) < L.
>>
>> By the standard covering lemma there exist x_1,...,
>> x_k in K and r_1, ..., r_k > 0 such that the balls
>> B(x_j,r_j) are disjoint, B(x_j, r_j) is contained
>> in V, and K is contained in the union of B(x_j,3r_j).
>> So
>>
>> � mu(K) <= sum mu(B(x_j,3r_j))
>>
>> � � � �<= L sum m(B(x_j,3r_j))
>>
>> � � � � = 3^n L sum m(B(x_j,r_j))
>>
>> � � � � = 3^n L m(union B(x_j,r_j))
>>
>> � � � � <= 3^n L m(V)
>>
>> � � � � < 3^n L epsilon.
>>
>> Let epsilon -> 0 and it follows that mu(K) = 0. QED.
>
>Looks correct to me, not that that ought to count for much.
I wonder why it's not in those books. Especially Rudin,
since he does consider the question of what happens
ae[mu].
Gc
> Suppose mu is a regular Borel measure on R^n. Define
>
> D mu(x) = lim_{r -> 0} mu(B(x,r)) / m(B(x,r))
>
> for all x for which the limit exists; here m is Lebesgue
> measure.
>
> Say mu is "singular" if it is singular with respect
> to m, ie mu and m are mutually singular (concentrated
> on disjoint sets).
>
> It's well known that if mu is singular then
>
> (*) D mu(x) = 0 ae[m].
>
> It also seems to be true that if mu is singular then
>
> (**) D mu(x) = infinity ae[mu].
Isn`t this just the Theorem 7.15 on Rudin 3th edition?
Gc
> On 12 marras, 13:23, David C. Ullrich <dullr...@sprynet.com> wrote:
>
>
>
> > Suppose mu is a regular Borel measure on R^n. Define
>
> > D mu(x) = lim_{r -> 0} mu(B(x,r)) / m(B(x,r))
>
> > for all x for which the limit exists; here m is Lebesgue
> > measure.
>
> > Say mu is "singular" if it is singular with respect
> > to m, ie mu and m are mutually singular (concentrated
> > on disjoint sets).
>
> > It's well known that if mu is singular then
>
> > (*) D mu(x) = 0 ae[m].
>
> > It also seems to be true that if mu is singular then
>
> > (**) D mu(x) = infinity ae[mu].
>
> Isn`t this just the Theorem 7.15 on Rudin 3th edition?
I mean we are talking about _positive_ measures.
Otherwise the proof below doesn`t obviously hold.
David C. Ullrich
<ebad5bad-5e17-4220...@h34g2000yqm.googlegroups.com>,
Gc <gcu...@hotmail.com> wrote:
> On 12 marras, 13:23, David C. Ullrich <dullr...@sprynet.com> wrote:
> > Suppose mu is a regular Borel measure on R^n. Define
> >
> > � D mu(x) = lim_{r -> 0} mu(B(x,r)) / m(B(x,r))
> >
> > for all x for which the limit exists; here m is Lebesgue
> > measure.
> >
> > Say mu is "singular" if it is singular with respect
> > to m, ie mu and m are mutually singular (concentrated
> > on disjoint sets).
> >
> > It's well known that if mu is singular then
> >
> > (*) � D mu(x) = 0 ae[m].
> >
> > It also seems to be true that if mu is singular then
> >
> > (**) �D mu(x) = infinity ae[mu].
>
> Isn`t this just the Theorem 7.15 on Rudin 3th edition?
Perhaps. My copy of the third edition is missing right now.
Previous editions definitely don't contain this result.
They contain a result that looks like this, but the definition
of D is different.
What's the definition of D in the book in front of you?
--
David C. Ullrich
Gc
> On 12 marras, 17:47, Gc <gcut...@hotmail.com> wrote:
>
>
>
> > On 12 marras, 13:23, David C. Ullrich <dullr...@sprynet.com> wrote:
>
> > > Suppose mu is a regular Borel measure on R^n. Define
>
> > > D mu(x) = lim_{r -> 0} mu(B(x,r)) / m(B(x,r))
>
> > > for all x for which the limit exists; here m is Lebesgue
> > > measure.
>
> > > Say mu is "singular" if it is singular with respect
> > > to m, ie mu and m are mutually singular (concentrated
> > > on disjoint sets).
>
> > > It's well known that if mu is singular then
>
> > > (*) D mu(x) = 0 ae[m].
>
> > > It also seems to be true that if mu is singular then
>
> > > (**) D mu(x) = infinity ae[mu].
>
> > Isn`t this just the Theorem 7.15 on Rudin 3th edition?
>
> I mean we are talking about _positive_ measures.
> Otherwise the proof below doesn`t obviously hold.
It also seems that the Rudin in his proof doens`t even suppose that
the measure is regular, only positive and Borel.
Gc
> In article
> <ebad5bad-5e17-4220-8c7c-33a0105b6...@h34g2000yqm.googlegroups.com>,
>
>
>
> Gc <gcut...@hotmail.com> wrote:
> > On 12 marras, 13:23, David C. Ullrich <dullr...@sprynet.com> wrote:
> > > Suppose mu is a regular Borel measure on R^n. Define
>
> > > D mu(x) = lim_{r -> 0} mu(B(x,r)) / m(B(x,r))
>
> > > for all x for which the limit exists; here m is Lebesgue
> > > measure.
>
> > > Say mu is "singular" if it is singular with respect
> > > to m, ie mu and m are mutually singular (concentrated
> > > on disjoint sets).
>
> > > It's well known that if mu is singular then
>
> > > (*) D mu(x) = 0 ae[m].
>
> > > It also seems to be true that if mu is singular then
>
> > > (**) D mu(x) = infinity ae[mu].
>
> > Isn`t this just the Theorem 7.15 on Rudin 3th edition?
>
> Perhaps. My copy of the third edition is missing right now.
> Previous editions definitely don't contain this result.
> They contain a result that looks like this, but the definition
> of D is different.
>
> What's the definition of D in the book in front of you?
It`s the ball definition. Dmu(x)= lim r-->o mu(B(x,r)/m(B(x,r). You
wouldn`t be much of criminal if you downloaded this book from the
rapidshare, because you own the book.
David C. Ullrich
wrote:
>On 12 marras, 18:46, "David C. Ullrich" <dullr...@sprynet.com> wrote:
>> In article
>> <ebad5bad-5e17-4220-8c7c-33a0105b6...@h34g2000yqm.googlegroups.com>,
>>
>>
>>
>> �Gc <gcut...@hotmail.com> wrote:
>> > On 12 marras, 13:23, David C. Ullrich <dullr...@sprynet.com> wrote:
>> > > Suppose mu is a regular Borel measure on R^n. Define
>>
>> > > � D mu(x) = lim_{r -> 0} mu(B(x,r)) / m(B(x,r))
>>
>> > > for all x for which the limit exists; here m is Lebesgue
>> > > measure.
>>
>> > > Say mu is "singular" if it is singular with respect
>> > > to m, ie mu and m are mutually singular (concentrated
>> > > on disjoint sets).
>>
>> > > It's well known that if mu is singular then
>>
>> > > (*) � D mu(x) = 0 ae[m].
>>
>> > > It also seems to be true that if mu is singular then
>>
>> > > (**) �D mu(x) = infinity ae[mu].
>>
>> > Isn`t this just the Theorem 7.15 on Rudin 3th edition?
Yes it is. I forgot how that chapter was totally different in the
third edition, should have checked that first.
>> Perhaps. My copy of the third edition is missing right now.
>> Previous editions definitely don't contain this result.
>> They contain a result that looks like this, but the definition
>> of D is different.
>>
>> What's the definition of D in the book in front of you?
>
>It`s the ball definition. Dmu(x)= lim r-->o mu(B(x,r)/m(B(x,r). You
>wouldn`t be much of criminal if you downloaded this book from the
>rapidshare, because you own the book.
Thanks. It was at home, as I suspected.
Rotwang
On 12 Nov, 11:23, David C. Ullrich <dullr...@sprynet.com> wrote:
>
> [...]
>
> The two books I know are Folland and Rudin.
By the way, which, if either, of those two books would you recommend
as a reference for measure and integration theory? I read Royden's
"Real Analysis" years ago, but recently while trying to read Rudin's
"Functional Analysis" I've found many gaps in my knowledge that need
to be filled. For example Royden only covers the Lebesgue measure on
R, and proves nothing about R^n, n > 1. Also the version of the Riesz
representation theorem he gives only involves regular Baire measures,
so I'd like to see a proof of the corresponding statement about
regular Borel measures. I'm after a book in which it would be easy to
find specific things I need without knowing the entire text back-to-
front (since I won't have time to read it from start to finish in the
near future). Any suggestions?
David C. Ullrich
<sg...@hotmail.co.uk> wrote:
>Threadjack...
>
>On 12 Nov, 11:23, David C. Ullrich <dullr...@sprynet.com> wrote:
>>
>> [...]
>>
>> The two books I know are Folland and Rudin.
>
>By the way, which, if either, of those two books would you recommend
>as a reference for measure and integration theory?
They're both excellent in my opinion.
I can't bring myself to endorse your plan of just looking things up
as needed, seems to me you should actually read one book at least
quickly. That might make Rudin a better choice because it's
shorter (since only half the book is on real analysis) while Folland
seems to me a little more encyclopedic. Also if the goal is to
get through Rudin's "Functional Analysis" it will probably happen
that Rudin's point of view and terminology, etc, will be better
for that.
Seems to me the biggest difference is that in Folland measures
come from some slightly hairy but very nice and very important
results of Caratheodory - in particular the Caratheodory stuff
about premeasures on algebras generating outer measures which
then lead to measures is used to get the Riesz Representation
Theorem later in the book. In Rudin the Riesz Representation
Theorem is done at the start, and then _it_ is where most
measures come from.
>I read Royden's
>"Real Analysis" years ago, but recently while trying to read Rudin's
>"Functional Analysis" I've found many gaps in my knowledge that need
>to be filled. For example Royden only covers the Lebesgue measure on
>R, and proves nothing about R^n, n > 1. Also the version of the Riesz
>representation theorem he gives only involves regular Baire measures,
>so I'd like to see a proof of the corresponding statement about
>regular Borel measures. I'm after a book in which it would be easy to
>find specific things I need without knowing the entire text back-to-
>front (since I won't have time to read it from start to finish in the
>near future). Any suggestions?
David C. Ullrich
Rotwang
> On Fri, 13 Nov 2009 21:17:20 -0800 (PST), Rotwang
>
> <sg...@hotmail.co.uk> wrote:
> >Threadjack...
>
> >On 12 Nov, 11:23, David C. Ullrich <dullr...@sprynet.com> wrote:
>
> >> [...]
>
> >> The two books I know are Folland and Rudin.
>
> >By the way, which, if either, of those two books would you recommend
> >as a reference for measure and integration theory?
>
> They're both excellent in my opinion.
>
> I can't bring myself to endorse your plan of just looking things up
> as needed, seems to me you should actually read one book at least
> quickly.
Whichever I get, I intend to read it at some point. It's just that I
have a lot of other work to do over the next few months.
> That might make Rudin a better choice because it's
> shorter (since only half the book is on real analysis) while Folland
> seems to me a little more encyclopedic. Also if the goal is to
> get through Rudin's "Functional Analysis" it will probably happen
> that Rudin's point of view and terminology, etc, will be better
> for that.
>
> Seems to me the biggest difference is that in Folland measures
> come from some slightly hairy but very nice and very important
> results of Caratheodory - in particular the Caratheodory stuff
> about premeasures on algebras generating outer measures which
> then lead to measures is used to get the Riesz Representation
> Theorem later in the book. In Rudin the Riesz Representation
> Theorem is done at the start, and then _it_ is where most
> measures come from.
Thanks.
G. A. Edgar
> >
> > The two books I know are Folland and Rudin.
Another one I like is
D. L. Cohn, MEASURE THEORY, Birkh�user
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Gc
wrote:
> > > The two books I know are Folland and Rudin.
>
> Another one I like is
Always good to know about new-to-me intresting new books. IMHO,
Folland`s book is a good book, but it has a swinish prize tag.
David Bernier
> On 15 marras, 12:42, "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
> wrote:
>>>> The two books I know are Folland and Rudin.
>> Another one I like is
>
> Always good to know about new-to-me intresting new books. IMHO,
> Folland`s book is a good book, but it has a swinish prize tag.
I have Cohn's book, the 1980 edition,
ISBN 3-7643-3003-1 .
Its main focus is measure and integration, and here's a list of Chapter
Titles:
1. Measures 2. Functions and Integrals
3. Convergence 4. Signed and Complex Measures
5. Product Measures 6. Differentiation
7. Measures on Locally Compact Spaces
8. Polish Spaces and Analytic Sets
9. Haar Measure
+ Appendices.
Includes Bibliography, Index of Notation and Index. (373 pp.).
The Bibliography mentions some well-known works.
I suppose suitability goes hand-in-hand with the degree of
in-depth coverage a reader is after. For instance,
I never ventured into Chapter 8 on Polish Sets & Analytic Sets.
OTOH, some topics in Functional Analysis may not be covered in
Cohn's book, e.g. the Krein�Milman theorem, cf.:
< http://en.wikipedia.org/wiki/Krein%E2%80%93Milman_theorem >
Theorem 7.2.8 states the Riesz representation theorem
for locally compact Hausdorff spaces, and positive linear
functionals. One part of the proof is
positive linear functional ==> the Borel measure is regular.
Also, many exercises and discussions outside the
"Definition, Proof, Theorem" terse type of presentation are
included.
I don't have the Folland Rudin texts handy, so
I'll refrain from commenting on them.
David Bernier
Gc
> Gc wrote:
> > On 15 marras, 12:42, "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
> > wrote:
> >>>> The two books I know are Folland and Rudin.
> >> Another one I like is
>
> > Always good to know about new-to-me intresting new books. IMHO,
> > Folland`s book is a good book, but it has a swinish prize tag.
>
> I have Cohn's book, the 1980 edition,
> ISBN 3-7643-3003-1 .
>
> Its main focus is measure and integration, and here's a list of Chapter
> Titles:
> 1. Measures 2. Functions and Integrals
> 3. Convergence 4. Signed and Complex Measures
> 5. Product Measures 6. Differentiation
>
> 7. Measures on Locally Compact Spaces
>
> 8. Polish Spaces and Analytic Sets
>
> 9. Haar Measure
> + Appendices.
> Includes Bibliography, Index of Notation and Index. (373 pp.).
> The Bibliography mentions some well-known works.
>
> I suppose suitability goes hand-in-hand with the degree of
> in-depth coverage a reader is after. For instance,
> I never ventured into Chapter 8 on Polish Sets & Analytic Sets.
>
> OTOH, some topics in Functional Analysis may not be covered in
> <http://en.wikipedia.org/wiki/Krein%E2%80%93Milman_theorem>
>
> Theorem 7.2.8 states the Riesz representation theorem
> for locally compact Hausdorff spaces, and positive linear
> functionals. One part of the proof is
> positive linear functional ==> the Borel measure is regular.
>
> Also, many exercises and discussions outside the
> "Definition, Proof, Theorem" terse type of presentation are
> included.
Thanks, this book looks very good. I`ll have good use for my tax
returns before Christmas.
G. A. Edgar
>
> I suppose suitability goes hand-in-hand with the degree of
> in-depth coverage a reader is after. For instance,
> I never ventured into Chapter 8 on Polish Sets & Analytic Sets.
>
In my opinion, a fine introduction to that subject. I have sometimes
used this chapter of Cohn to supplement other textbooks that don't have
this material.
David C. Ullrich
<davi...@videotron.ca> wrote:
>Gc wrote:
>> On 15 marras, 12:42, "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
>> wrote:
>>>>> The two books I know are Folland and Rudin.
>>> Another one I like is
>>> D. L. Cohn, MEASURE THEORY, Birkh�user
>>
>> Always good to know about new-to-me intresting new books. IMHO,
>> Folland`s book is a good book, but it has a swinish prize tag.
>
>I have Cohn's book, the 1980 edition,
>ISBN 3-7643-3003-1 .
>
>Its main focus is measure and integration, and here's a list of Chapter
>Titles:
>1. Measures 2. Functions and Integrals
>3. Convergence 4. Signed and Complex Measures
>5. Product Measures 6. Differentiation
>
>7. Measures on Locally Compact Spaces
>
>8. Polish Spaces and Analytic Sets
>
>9. Haar Measure
>+ Appendices.
>Includes Bibliography, Index of Notation and Index. (373 pp.).
>The Bibliography mentions some well-known works.
>
>I suppose suitability goes hand-in-hand with the degree of
>in-depth coverage a reader is after. For instance,
>I never ventured into Chapter 8 on Polish Sets & Analytic Sets.
>
>OTOH, some topics in Functional Analysis may not be covered in
>Cohn's book, e.g. the Krein�Milman theorem, cf.:
>< http://en.wikipedia.org/wiki/Krein%E2%80%93Milman_theorem >
This is more or less what one would expect from the title.
"Real analysis" is not the same thing as "measure theory",
although of course measure theory is a huge part of
real analysis. One would expect a book titled
"Measure Theory" to go into more depth regarding
measure theory and hence have less space for the
rest of real analysis.
>Theorem 7.2.8 states the Riesz representation theorem
>for locally compact Hausdorff spaces, and positive linear
>functionals. One part of the proof is
>positive linear functional ==> the Borel measure is regular.
>
>Also, many exercises and discussions outside the
>"Definition, Proof, Theorem" terse type of presentation are
>included.
>
>I don't have the Folland Rudin texts handy, so
>I'll refrain from commenting on them.
>
>David Bernier
David C. Ullrich