simultanious convergence of integral norms
$begingroup$
Suppose I have a sequence of measurable functions $f_jin L^pcap L^q$. I am wondering what can I deduce about the behavior of the sequence in one space based on its behavior in the other.
Specifically, I am interested in the following two questions:
- If $f_j$ is cauchy with respect to the norm on $L^p$ is it cauchy with respect to the $L^q$ norm as well?
- If $f_j$ converges in $L^p$ to $g$ and in $L^q$ to $h$, is it true that $g=h$ almost everywhere?
Similar to this question but I am not necessarily interested in the case of $mu(X)<infty$
convergence metric-spaces lebesgue-integral lp-spaces
asked Dec 23 '18 at 21:40
Bar AlonBar Alon
504115
$endgroup$
add a comment |
$begingroup$
Suppose I have a sequence of measurable functions $f_jin L^pcap L^q$. I am wondering what can I deduce about the behavior of the sequence in one space based on its behavior in the other.
Specifically, I am interested in the following two questions:
- If $f_j$ is cauchy with respect to the norm on $L^p$ is it cauchy with respect to the $L^q$ norm as well?
- If $f_j$ converges in $L^p$ to $g$ and in $L^q$ to $h$, is it true that $g=h$ almost everywhere?
Similar to this question but I am not necessarily interested in the case of $mu(X)<infty$
convergence metric-spaces lebesgue-integral lp-spaces
asked Dec 23 '18 at 21:40
Bar AlonBar Alon
504115
$endgroup$
add a comment |
$begingroup$
Suppose I have a sequence of measurable functions $f_jin L^pcap L^q$. I am wondering what can I deduce about the behavior of the sequence in one space based on its behavior in the other.
Specifically, I am interested in the following two questions:
- If $f_j$ is cauchy with respect to the norm on $L^p$ is it cauchy with respect to the $L^q$ norm as well?
- If $f_j$ converges in $L^p$ to $g$ and in $L^q$ to $h$, is it true that $g=h$ almost everywhere?
Similar to this question but I am not necessarily interested in the case of $mu(X)<infty$
convergence metric-spaces lebesgue-integral lp-spaces
asked Dec 23 '18 at 21:40
Bar AlonBar Alon
504115
$endgroup$
Suppose I have a sequence of measurable functions $f_jin L^pcap L^q$. I am wondering what can I deduce about the behavior of the sequence in one space based on its behavior in the other.
Specifically, I am interested in the following two questions:
- If $f_j$ is cauchy with respect to the norm on $L^p$ is it cauchy with respect to the $L^q$ norm as well?
- If $f_j$ converges in $L^p$ to $g$ and in $L^q$ to $h$, is it true that $g=h$ almost everywhere?
Similar to this question but I am not necessarily interested in the case of $mu(X)<infty$
convergence metric-spaces lebesgue-integral lp-spaces
convergence metric-spaces lebesgue-integral lp-spaces
asked Dec 23 '18 at 21:40
Bar AlonBar Alon
504115
asked Dec 23 '18 at 21:40
Bar AlonBar Alon
504115
asked Dec 23 '18 at 21:40
Bar AlonBar Alon
504115
asked Dec 23 '18 at 21:40
Bar AlonBar Alon
504115
asked Dec 23 '18 at 21:40
Bar AlonBar Alon
504115
504115
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Whether convergence in one $L^p$ implies convergence in another one depends on the ground space on which the functions are defined. The case of finite measure is explained by Hölder's inequality and the case of a purely atomic space is the other way round.
For the other question: convergence in any $L^p$ implies convergence in measure which implies that the limits have to be the same.
answered Dec 23 '18 at 22:31
DirkDirk
8,8102447
$endgroup$
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%2f3050725%2fsimultanious-convergence-of-integral-norms%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
$begingroup$
Whether convergence in one $L^p$ implies convergence in another one depends on the ground space on which the functions are defined. The case of finite measure is explained by Hölder's inequality and the case of a purely atomic space is the other way round.
For the other question: convergence in any $L^p$ implies convergence in measure which implies that the limits have to be the same.
answered Dec 23 '18 at 22:31
DirkDirk
8,8102447
$endgroup$
add a comment |
$begingroup$
Whether convergence in one $L^p$ implies convergence in another one depends on the ground space on which the functions are defined. The case of finite measure is explained by Hölder's inequality and the case of a purely atomic space is the other way round.
For the other question: convergence in any $L^p$ implies convergence in measure which implies that the limits have to be the same.
answered Dec 23 '18 at 22:31
DirkDirk
8,8102447
$endgroup$
add a comment |
$begingroup$
Whether convergence in one $L^p$ implies convergence in another one depends on the ground space on which the functions are defined. The case of finite measure is explained by Hölder's inequality and the case of a purely atomic space is the other way round.
For the other question: convergence in any $L^p$ implies convergence in measure which implies that the limits have to be the same.
answered Dec 23 '18 at 22:31
DirkDirk
8,8102447
$endgroup$
Whether convergence in one $L^p$ implies convergence in another one depends on the ground space on which the functions are defined. The case of finite measure is explained by Hölder's inequality and the case of a purely atomic space is the other way round.
For the other question: convergence in any $L^p$ implies convergence in measure which implies that the limits have to be the same.
answered Dec 23 '18 at 22:31
DirkDirk
8,8102447
answered Dec 23 '18 at 22:31
DirkDirk
8,8102447
answered Dec 23 '18 at 22:31
DirkDirk
8,8102447
answered Dec 23 '18 at 22:31
DirkDirk
8,8102447
8,8102447
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%2f3050725%2fsimultanious-convergence-of-integral-norms%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
