Dgdrxrt

Homework question about an engine pulling a train

How to handle characters who are more educated than the author?

Am I ethically obligated to go into work on an off day if the reason is sudden?

How to support a colleague who finds meetings extremely tiring?

60's-70's movie: home appliances revolting against the owners

Keeping a retro style to sci-fi spaceships?

Was credit for the black hole image misappropriated?

How do you keep chess fun when your opponent constantly beats you?

How did the audience guess the pentatonic scale in Bobby McFerrin's presentation?

Is this wall load bearing? Blueprints and photos attached

How do I design a circuit to convert a 100 mV and 50 Hz sine wave to a square wave?

Match Roman Numerals

Simulating Exploding Dice

How to type a long/em dash `—`

Why are PDP-7-style microprogrammed instructions out of vogue?

Does Parliament need to approve the new Brexit delay to 31 October 2019?

Why not take a picture of a closer black hole?

Are there continuous functions who are the same in an interval but differ in at least one other point?

Is it ok to offer lower paid work as a trial period before negotiating for a full-time job?

US Healthcare consultation for visitors

Does Parliament hold absolute power in the UK?

Is there a writing software that you can sort scenes like slides in PowerPoint?

What to do when moving next to a bird sanctuary with a loosely-domesticated cat?

Huge performance difference of the command find with and without using %M option to show permissions

Hahn Decomposition Theorem In Folland

4

$begingroup$

I was reading the proof of Hahn Decomposition theorem from the textbook of Folland: precisely I was looking at the following text

I have the following question:

1. As Highlighted in the text above, why $m$ is finite? It may be infinite as there is no restriction on $X$. Why does the author consider it finite?


2. Again why $nu(A)<infty$ ? I do not understand also this.


3. I understand that even for $A$ also we get some $B$ with the property that $nu(B)>nu(A)+1/n$ but I do not understand how this leads to a contradiction.

I would be really thankful if someone could help me. Any help will be appreciated.

real-analysis measure-theory proof-explanation lebesgue-measure

share|cite|improve this question

edited Mar 31 at 9:50

dmtri

1,7612521

asked Mar 31 at 8:28

SRJSRJ

1,8981620

$endgroup$

add a comment | 

4

$begingroup$

I was reading the proof of Hahn Decomposition theorem from the textbook of Folland: precisely I was looking at the following text

I have the following question:

1. As Highlighted in the text above, why $m$ is finite? It may be infinite as there is no restriction on $X$. Why does the author consider it finite?


2. Again why $nu(A)<infty$ ? I do not understand also this.


3. I understand that even for $A$ also we get some $B$ with the property that $nu(B)>nu(A)+1/n$ but I do not understand how this leads to a contradiction.

I would be really thankful if someone could help me. Any help will be appreciated.

real-analysis measure-theory proof-explanation lebesgue-measure

share|cite|improve this question

edited Mar 31 at 9:50

dmtri

1,7612521

asked Mar 31 at 8:28

SRJSRJ

1,8981620

$endgroup$

add a comment | 

$begingroup$

I was reading the proof of Hahn Decomposition theorem from the textbook of Folland: precisely I was looking at the following text

I have the following question:

1. As Highlighted in the text above, why $m$ is finite? It may be infinite as there is no restriction on $X$. Why does the author consider it finite?


2. Again why $nu(A)<infty$ ? I do not understand also this.


3. I understand that even for $A$ also we get some $B$ with the property that $nu(B)>nu(A)+1/n$ but I do not understand how this leads to a contradiction.

I would be really thankful if someone could help me. Any help will be appreciated.

real-analysis measure-theory proof-explanation lebesgue-measure

share|cite|improve this question

edited Mar 31 at 9:50

dmtri

1,7612521

asked Mar 31 at 8:28

SRJSRJ

1,8981620

$endgroup$

share|cite|improve this question

edited Mar 31 at 9:50

dmtri

1,7612521

asked Mar 31 at 8:28

SRJSRJ

1,8981620

share|cite|improve this question

edited Mar 31 at 9:50

dmtri

1,7612521

asked Mar 31 at 8:28

SRJSRJ

1,8981620

edited Mar 31 at 9:50

dmtri

1,7612521

edited Mar 31 at 9:50

dmtri

1,7612521

asked Mar 31 at 8:28

SRJSRJ

1,8981620

asked Mar 31 at 8:28

SRJSRJ

1,8981620

1 Answer
1

active

oldest

votes

0

$begingroup$

Perhaps if you consult Folland's definition of "signed measure" you will find that all values are finite. $nu : mathcal X to (-infty,+infty)$.

According to Wikipedia:

There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. In research papers and advanced books signed measures are usually only allowed to take finite values, while undergraduate textbooks often allow them to take infinite values.

Suppose we allow infinite values. Then Lebesgue measure on $mathbb R$ would be a finite measure, $E = mathbb R$ would be a positive set, and $nu(E)=infty$ on that case.

share|cite|improve this answer

edited Mar 31 at 10:59

answered Mar 31 at 10:37

GEdgarGEdgar

63.5k269175

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Did you look this up? Signed measures can take infinite values as well.
  $endgroup$
  – user370967
  Mar 31 at 10:45
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @GEdgar Dear Sir , Signed measure can take at most on infinite value either positive or negative.
  $endgroup$
  – SRJ
  Mar 31 at 10:57
  
  

  
  
  
  

add a comment | 

Your Answer

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3169166%2fhahn-decomposition-theorem-in-folland%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

0

$begingroup$

Perhaps if you consult Folland's definition of "signed measure" you will find that all values are finite. $nu : mathcal X to (-infty,+infty)$.

According to Wikipedia:

There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. In research papers and advanced books signed measures are usually only allowed to take finite values, while undergraduate textbooks often allow them to take infinite values.

Suppose we allow infinite values. Then Lebesgue measure on $mathbb R$ would be a finite measure, $E = mathbb R$ would be a positive set, and $nu(E)=infty$ on that case.

share|cite|improve this answer

edited Mar 31 at 10:59

answered Mar 31 at 10:37

GEdgarGEdgar

63.5k269175

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Did you look this up? Signed measures can take infinite values as well.
  $endgroup$
  – user370967
  Mar 31 at 10:45
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @GEdgar Dear Sir , Signed measure can take at most on infinite value either positive or negative.
  $endgroup$
  – SRJ
  Mar 31 at 10:57
  
  

  
  
  
  

add a comment | 

0

$begingroup$

Perhaps if you consult Folland's definition of "signed measure" you will find that all values are finite. $nu : mathcal X to (-infty,+infty)$.

According to Wikipedia:

There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. In research papers and advanced books signed measures are usually only allowed to take finite values, while undergraduate textbooks often allow them to take infinite values.

Suppose we allow infinite values. Then Lebesgue measure on $mathbb R$ would be a finite measure, $E = mathbb R$ would be a positive set, and $nu(E)=infty$ on that case.

share|cite|improve this answer

edited Mar 31 at 10:59

answered Mar 31 at 10:37

GEdgarGEdgar

63.5k269175

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Did you look this up? Signed measures can take infinite values as well.
  $endgroup$
  – user370967
  Mar 31 at 10:45
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @GEdgar Dear Sir , Signed measure can take at most on infinite value either positive or negative.
  $endgroup$
  – SRJ
  Mar 31 at 10:57
  
  

  
  
  
  

add a comment | 

$begingroup$

Perhaps if you consult Folland's definition of "signed measure" you will find that all values are finite. $nu : mathcal X to (-infty,+infty)$.

According to Wikipedia:

There are two slightly different concepts of a signed measure, depending on whether or not one allows it to take infinite values. In research papers and advanced books signed measures are usually only allowed to take finite values, while undergraduate textbooks often allow them to take infinite values.

Suppose we allow infinite values. Then Lebesgue measure on $mathbb R$ would be a finite measure, $E = mathbb R$ would be a positive set, and $nu(E)=infty$ on that case.

share|cite|improve this answer

edited Mar 31 at 10:59

answered Mar 31 at 10:37

GEdgarGEdgar

63.5k269175

$endgroup$

share|cite|improve this answer

edited Mar 31 at 10:59

answered Mar 31 at 10:37

GEdgarGEdgar

63.5k269175

answered Mar 31 at 10:37

GEdgarGEdgar

63.5k269175

answered Mar 31 at 10:37

GEdgarGEdgar

63.5k269175