$ limsup_{nrightarrowinfty}int_{Omega_varepsilon} f_nleqint_{Omega_varepsilon} f $ implies that of $Omega$?
$begingroup$
Let $f_n,fin L^1(Omega)$, all non-negative. Let $Omega_varepsilon$ be an increasing set sequence that $bigcup_{varepsilon>0}Omega_varepsilon=Omega$. If for each $varepsilon$,
$
limsup_{nrightarrowinfty}int_{Omega_varepsilon} f_nleqint_{Omega_varepsilon} f
$
Do we have
$
limsup_{nrightarrowinfty}int_{Omega} f_nleqint_{Omega} f
?$
real-analysis
asked Dec 18 '18 at 9:29
P.RP.R
184
$endgroup$
add a comment |
$begingroup$
Let $f_n,fin L^1(Omega)$, all non-negative. Let $Omega_varepsilon$ be an increasing set sequence that $bigcup_{varepsilon>0}Omega_varepsilon=Omega$. If for each $varepsilon$,
$
limsup_{nrightarrowinfty}int_{Omega_varepsilon} f_nleqint_{Omega_varepsilon} f
$
Do we have
$
limsup_{nrightarrowinfty}int_{Omega} f_nleqint_{Omega} f
?$
real-analysis
asked Dec 18 '18 at 9:29
P.RP.R
184
$endgroup$
$begingroup$
What kind of assumptions do you have on $Omega$? Is it a bounded open set?
$endgroup$
– BigbearZzz
Dec 18 '18 at 9:36
add a comment |
$begingroup$
Let $f_n,fin L^1(Omega)$, all non-negative. Let $Omega_varepsilon$ be an increasing set sequence that $bigcup_{varepsilon>0}Omega_varepsilon=Omega$. If for each $varepsilon$,
$
limsup_{nrightarrowinfty}int_{Omega_varepsilon} f_nleqint_{Omega_varepsilon} f
$
Do we have
$
limsup_{nrightarrowinfty}int_{Omega} f_nleqint_{Omega} f
?$
real-analysis
asked Dec 18 '18 at 9:29
P.RP.R
184
$endgroup$
Let $f_n,fin L^1(Omega)$, all non-negative. Let $Omega_varepsilon$ be an increasing set sequence that $bigcup_{varepsilon>0}Omega_varepsilon=Omega$. If for each $varepsilon$,
$
limsup_{nrightarrowinfty}int_{Omega_varepsilon} f_nleqint_{Omega_varepsilon} f
$
Do we have
$
limsup_{nrightarrowinfty}int_{Omega} f_nleqint_{Omega} f
?$
real-analysis
real-analysis
asked Dec 18 '18 at 9:29
P.RP.R
184
asked Dec 18 '18 at 9:29
P.RP.R
184
asked Dec 18 '18 at 9:29
P.RP.R
184
asked Dec 18 '18 at 9:29
P.RP.R
184
asked Dec 18 '18 at 9:29
P.RP.R
184
184
$begingroup$
What kind of assumptions do you have on $Omega$? Is it a bounded open set?
$endgroup$
– BigbearZzz
Dec 18 '18 at 9:36
add a comment |
$begingroup$
What kind of assumptions do you have on $Omega$? Is it a bounded open set?
$endgroup$
– BigbearZzz
Dec 18 '18 at 9:36
$begingroup$
What kind of assumptions do you have on $Omega$? Is it a bounded open set?
$endgroup$
– BigbearZzz
Dec 18 '18 at 9:36
$begingroup$
What kind of assumptions do you have on $Omega$? Is it a bounded open set?
$endgroup$
– BigbearZzz
Dec 18 '18 at 9:36
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No. Let $Omega =(0,1)$ with Lebesgue measure and $Omega_{epsilon} =(epsilon ,1)$. Let $f_n=nI_{(0,frac 1 n)}$, $f=0$. For each $epsilon$ we have $int_{Omega_{epsilon}} f_n=0$ as soon as $frac 1 n <epsilon$ but $int_{Omega} f_n=1$ for all $n$.
answered Dec 18 '18 at 9:37
Kavi Rama MurthyKavi Rama Murthy
60k42161
$endgroup$
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',
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
});
}
});
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3044946%2flimsup-n-rightarrow-infty-int-omega-varepsilon-f-n-leq-int-omega-var%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No. Let $Omega =(0,1)$ with Lebesgue measure and $Omega_{epsilon} =(epsilon ,1)$. Let $f_n=nI_{(0,frac 1 n)}$, $f=0$. For each $epsilon$ we have $int_{Omega_{epsilon}} f_n=0$ as soon as $frac 1 n <epsilon$ but $int_{Omega} f_n=1$ for all $n$.
answered Dec 18 '18 at 9:37
Kavi Rama MurthyKavi Rama Murthy
60k42161
$endgroup$
add a comment |
$begingroup$
No. Let $Omega =(0,1)$ with Lebesgue measure and $Omega_{epsilon} =(epsilon ,1)$. Let $f_n=nI_{(0,frac 1 n)}$, $f=0$. For each $epsilon$ we have $int_{Omega_{epsilon}} f_n=0$ as soon as $frac 1 n <epsilon$ but $int_{Omega} f_n=1$ for all $n$.
answered Dec 18 '18 at 9:37
Kavi Rama MurthyKavi Rama Murthy
60k42161
$endgroup$
add a comment |
$begingroup$
No. Let $Omega =(0,1)$ with Lebesgue measure and $Omega_{epsilon} =(epsilon ,1)$. Let $f_n=nI_{(0,frac 1 n)}$, $f=0$. For each $epsilon$ we have $int_{Omega_{epsilon}} f_n=0$ as soon as $frac 1 n <epsilon$ but $int_{Omega} f_n=1$ for all $n$.
answered Dec 18 '18 at 9:37
Kavi Rama MurthyKavi Rama Murthy
60k42161
$endgroup$
No. Let $Omega =(0,1)$ with Lebesgue measure and $Omega_{epsilon} =(epsilon ,1)$. Let $f_n=nI_{(0,frac 1 n)}$, $f=0$. For each $epsilon$ we have $int_{Omega_{epsilon}} f_n=0$ as soon as $frac 1 n <epsilon$ but $int_{Omega} f_n=1$ for all $n$.
answered Dec 18 '18 at 9:37
Kavi Rama MurthyKavi Rama Murthy
60k42161
answered Dec 18 '18 at 9:37
Kavi Rama MurthyKavi Rama Murthy
60k42161
answered Dec 18 '18 at 9:37
Kavi Rama MurthyKavi Rama Murthy
60k42161
answered Dec 18 '18 at 9:37
Kavi Rama MurthyKavi Rama Murthy
60k42161
60k42161
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3044946%2flimsup-n-rightarrow-infty-int-omega-varepsilon-f-n-leq-int-omega-var%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
What kind of assumptions do you have on $Omega$? Is it a bounded open set?
$endgroup$
– BigbearZzz
Dec 18 '18 at 9:36