In $mathbb R^n$, why does $f_k=chi_mathbf x$ not converge globally in measure to $f=1$?

Clash Royale CLAN TAG#URR8PPP

up vote
0
down vote

favorite

In $mathbb R^n$, why does $f_k=chi_mathbf x: $ not converge globally in measure to $f=1$? I think as $kto +infty$, $f_k = 1$ in a ball of radius $+infty$ centered at $0$.

real-analysis

share|cite|improve this question

asked Jul 16 at 21:04

user398843

394215

add a comment | 

up vote
0
down vote

favorite

In $mathbb R^n$, why does $f_k=chi_mathbf x: $ not converge globally in measure to $f=1$? I think as $kto +infty$, $f_k = 1$ in a ball of radius $+infty$ centered at $0$.

real-analysis

share|cite|improve this question

asked Jul 16 at 21:04

user398843

394215

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

In $mathbb R^n$, why does $f_k=chi_mathbf x: $ not converge globally in measure to $f=1$? I think as $kto +infty$, $f_k = 1$ in a ball of radius $+infty$ centered at $0$.

real-analysis

share|cite|improve this question

asked Jul 16 at 21:04

user398843

394215

In $mathbb R^n$, why does $f_k=chi_mathbf x: $ not converge globally in measure to $f=1$? I think as $kto +infty$, $f_k = 1$ in a ball of radius $+infty$ centered at $0$.

real-analysis

share|cite|improve this question

asked Jul 16 at 21:04

user398843

394215

share|cite|improve this question

share|cite|improve this question

asked Jul 16 at 21:04

user398843

394215

asked Jul 16 at 21:04

user398843

394215

asked Jul 16 at 21:04

user398843

394215

394215

add a comment | 

add a comment | 

1 Answer
1

active

oldest

votes

up vote
0
down vote



accepted

Because for any $1>epsilon>0$ and any $k$:
$$mu(f-f_k) = mu(mathbbR^n)-mu(xinmathbbR^n: ) = infty $$
As $mu(xinmathbbR^n: ) = mu(x)<infty$

Where $mu$ is Lebesgue's measure.

share|cite|improve this answer

edited Jul 16 at 21:16

answered Jul 16 at 21:09

Rumpelstiltskin

1,424315

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Then we take the limit?
  – user398843
  Jul 16 at 21:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user398843 If you take limit of infinities, what do you get? Infinity.
  – Rumpelstiltskin
  Jul 16 at 21:17
  
  

  
  
  
  

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

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',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
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

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2853855%2fin-mathbb-rn-why-does-f-k-chi-mathbf-x-mathbf-x-k-not-conve%23new-answer', 'question_page');

);

Post as a guest

Name Email

1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote



accepted

Because for any $1>epsilon>0$ and any $k$:
$$mu(f-f_k) = mu(mathbbR^n)-mu(xinmathbbR^n: ) = infty $$
As $mu(xinmathbbR^n: ) = mu(x)<infty$

Where $mu$ is Lebesgue's measure.

share|cite|improve this answer

edited Jul 16 at 21:16

answered Jul 16 at 21:09

Rumpelstiltskin

1,424315

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Then we take the limit?
  – user398843
  Jul 16 at 21:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user398843 If you take limit of infinities, what do you get? Infinity.
  – Rumpelstiltskin
  Jul 16 at 21:17
  
  

  
  
  
  

add a comment | 

up vote
0
down vote



accepted

Because for any $1>epsilon>0$ and any $k$:
$$mu(f-f_k) = mu(mathbbR^n)-mu(xinmathbbR^n: ) = infty $$
As $mu(xinmathbbR^n: ) = mu(x)<infty$

Where $mu$ is Lebesgue's measure.

share|cite|improve this answer

edited Jul 16 at 21:16

answered Jul 16 at 21:09

Rumpelstiltskin

1,424315

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Then we take the limit?
  – user398843
  Jul 16 at 21:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user398843 If you take limit of infinities, what do you get? Infinity.
  – Rumpelstiltskin
  Jul 16 at 21:17
  
  

  
  
  
  

add a comment | 

up vote
0
down vote



accepted

up vote
0
down vote



accepted

Because for any $1>epsilon>0$ and any $k$:
$$mu(f-f_k) = mu(mathbbR^n)-mu(xinmathbbR^n: ) = infty $$
As $mu(xinmathbbR^n: ) = mu(x)<infty$

Where $mu$ is Lebesgue's measure.

share|cite|improve this answer

edited Jul 16 at 21:16

answered Jul 16 at 21:09

Rumpelstiltskin

1,424315

Because for any $1>epsilon>0$ and any $k$:
$$mu(f-f_k) = mu(mathbbR^n)-mu(xinmathbbR^n: ) = infty $$
As $mu(xinmathbbR^n: ) = mu(x)<infty$

Where $mu$ is Lebesgue's measure.

share|cite|improve this answer

edited Jul 16 at 21:16

answered Jul 16 at 21:09

Rumpelstiltskin

1,424315

share|cite|improve this answer

share|cite|improve this answer

edited Jul 16 at 21:16

edited Jul 16 at 21:16

edited Jul 16 at 21:16

answered Jul 16 at 21:09

Rumpelstiltskin

1,424315

answered Jul 16 at 21:09

Rumpelstiltskin

1,424315

answered Jul 16 at 21:09

Rumpelstiltskin

1,424315

1,424315

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Then we take the limit?
  – user398843
  Jul 16 at 21:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user398843 If you take limit of infinities, what do you get? Infinity.
  – Rumpelstiltskin
  Jul 16 at 21:17
  
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  

  
  
  
  
  
  

  
  Then we take the limit?
  – user398843
  Jul 16 at 21:17
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @user398843 If you take limit of infinities, what do you get? Infinity.
  – Rumpelstiltskin
  Jul 16 at 21:17
  
  

  
  
  
  

Then we take the limit?
– user398843
Jul 16 at 21:17

Then we take the limit?
– user398843
Jul 16 at 21:17

@user398843 If you take limit of infinities, what do you get? Infinity.
– Rumpelstiltskin
Jul 16 at 21:17

@user398843 If you take limit of infinities, what do you get? Infinity.
– Rumpelstiltskin
Jul 16 at 21:17

add a comment | 

 

draft saved

draft discarded

 

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

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2853855%2fin-mathbb-rn-why-does-f-k-chi-mathbf-x-mathbf-x-k-not-conve%23new-answer', 'question_page');

);

Post as a guest

Name Email

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

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

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

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Name Email

Name

Email

Name Email

Name

Email