$L^p$ integrability question
$begingroup$
The question is simply: If I have two random variables $X,Y in bigcap_{pgeq 1} L^p(Omega)$ and a third $Z$ r.v. such that
$$XZ=Y$$
can I conclude that $Zin bigcap_{pgeq 1}L^p(Omega)$?
(I know this would't hold for a single $L^p(Omega)$ space, the point is that both $X$ and $Y$ are in all $L^p(Omega)$, $pgeq 1$)
Or, in other words, if $Xneq 0$, $Xin bigcap_{pgeq 1} L^p(Omega)$ is then $1/X in bigcap_{pgeq 1} L^p(Omega)$?
Thank you very much for the help!
real-analysis probability-theory measure-theory random-variables lp-spaces
edited Dec 7 '18 at 13:47
Davide Giraudo
126k16150261
asked Aug 15 '13 at 13:29
MarkMark
82
$endgroup$
add a comment |
$begingroup$
The question is simply: If I have two random variables $X,Y in bigcap_{pgeq 1} L^p(Omega)$ and a third $Z$ r.v. such that
$$XZ=Y$$
can I conclude that $Zin bigcap_{pgeq 1}L^p(Omega)$?
(I know this would't hold for a single $L^p(Omega)$ space, the point is that both $X$ and $Y$ are in all $L^p(Omega)$, $pgeq 1$)
Or, in other words, if $Xneq 0$, $Xin bigcap_{pgeq 1} L^p(Omega)$ is then $1/X in bigcap_{pgeq 1} L^p(Omega)$?
Thank you very much for the help!
real-analysis probability-theory measure-theory random-variables lp-spaces
edited Dec 7 '18 at 13:47
Davide Giraudo
126k16150261
asked Aug 15 '13 at 13:29
MarkMark
82
$endgroup$
add a comment |
$begingroup$
The question is simply: If I have two random variables $X,Y in bigcap_{pgeq 1} L^p(Omega)$ and a third $Z$ r.v. such that
$$XZ=Y$$
can I conclude that $Zin bigcap_{pgeq 1}L^p(Omega)$?
(I know this would't hold for a single $L^p(Omega)$ space, the point is that both $X$ and $Y$ are in all $L^p(Omega)$, $pgeq 1$)
Or, in other words, if $Xneq 0$, $Xin bigcap_{pgeq 1} L^p(Omega)$ is then $1/X in bigcap_{pgeq 1} L^p(Omega)$?
Thank you very much for the help!
real-analysis probability-theory measure-theory random-variables lp-spaces
edited Dec 7 '18 at 13:47
Davide Giraudo
126k16150261
asked Aug 15 '13 at 13:29
MarkMark
82
$endgroup$
The question is simply: If I have two random variables $X,Y in bigcap_{pgeq 1} L^p(Omega)$ and a third $Z$ r.v. such that
$$XZ=Y$$
can I conclude that $Zin bigcap_{pgeq 1}L^p(Omega)$?
(I know this would't hold for a single $L^p(Omega)$ space, the point is that both $X$ and $Y$ are in all $L^p(Omega)$, $pgeq 1$)
Or, in other words, if $Xneq 0$, $Xin bigcap_{pgeq 1} L^p(Omega)$ is then $1/X in bigcap_{pgeq 1} L^p(Omega)$?
Thank you very much for the help!
real-analysis probability-theory measure-theory random-variables lp-spaces
real-analysis probability-theory measure-theory random-variables lp-spaces
edited Dec 7 '18 at 13:47
Davide Giraudo
126k16150261
asked Aug 15 '13 at 13:29
MarkMark
82
edited Dec 7 '18 at 13:47
Davide Giraudo
126k16150261
asked Aug 15 '13 at 13:29
MarkMark
82
edited Dec 7 '18 at 13:47
Davide Giraudo
126k16150261
edited Dec 7 '18 at 13:47
Davide Giraudo
126k16150261
edited Dec 7 '18 at 13:47
Davide Giraudo
126k16150261
126k16150261
asked Aug 15 '13 at 13:29
MarkMark
82
asked Aug 15 '13 at 13:29
MarkMark
82
asked Aug 15 '13 at 13:29
MarkMark
82
82
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Take $Omega:=(0,1)$ with Lebesgue measure and $X(omega):=omega$. $X$ is bounded, hence in all the $L^p$ (as $Omega$ has finite measure) but its reciprocal is not even integrable.
edited Dec 7 '18 at 5:53
Drew Brady
712315
answered Aug 15 '13 at 13:55
Davide GiraudoDavide Giraudo
126k16150261
$endgroup$
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%2f468267%2flp-integrability-question%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
$begingroup$
Take $Omega:=(0,1)$ with Lebesgue measure and $X(omega):=omega$. $X$ is bounded, hence in all the $L^p$ (as $Omega$ has finite measure) but its reciprocal is not even integrable.
edited Dec 7 '18 at 5:53
Drew Brady
712315
answered Aug 15 '13 at 13:55
Davide GiraudoDavide Giraudo
126k16150261
$endgroup$
add a comment |
$begingroup$
Take $Omega:=(0,1)$ with Lebesgue measure and $X(omega):=omega$. $X$ is bounded, hence in all the $L^p$ (as $Omega$ has finite measure) but its reciprocal is not even integrable.
edited Dec 7 '18 at 5:53
Drew Brady
712315
answered Aug 15 '13 at 13:55
Davide GiraudoDavide Giraudo
126k16150261
$endgroup$
add a comment |
$begingroup$
Take $Omega:=(0,1)$ with Lebesgue measure and $X(omega):=omega$. $X$ is bounded, hence in all the $L^p$ (as $Omega$ has finite measure) but its reciprocal is not even integrable.
edited Dec 7 '18 at 5:53
Drew Brady
712315
answered Aug 15 '13 at 13:55
Davide GiraudoDavide Giraudo
126k16150261
$endgroup$
Take $Omega:=(0,1)$ with Lebesgue measure and $X(omega):=omega$. $X$ is bounded, hence in all the $L^p$ (as $Omega$ has finite measure) but its reciprocal is not even integrable.
edited Dec 7 '18 at 5:53
Drew Brady
712315
answered Aug 15 '13 at 13:55
Davide GiraudoDavide Giraudo
126k16150261
edited Dec 7 '18 at 5:53
Drew Brady
712315
edited Dec 7 '18 at 5:53
Drew Brady
712315
edited Dec 7 '18 at 5:53
Drew Brady
712315
712315
answered Aug 15 '13 at 13:55
Davide GiraudoDavide Giraudo
126k16150261
answered Aug 15 '13 at 13:55
Davide GiraudoDavide Giraudo
126k16150261
answered Aug 15 '13 at 13:55
Davide GiraudoDavide Giraudo
126k16150261
126k16150261
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%2f468267%2flp-integrability-question%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
