Switching improper integrals without Fubini
$begingroup$
I'm trying to understand general conditions that permit switching integrals as in
$$int_a^infty int_a^infty f(x,y) dx dy = int_a^infty int_a^infty f(x,y) dy dx $$
if $f$ is not nonnegative or nonpositive and the integrals $int_a^infty f(x,y)dx, int_a^infty f(x,y)dy$ are improper Riemann integrals that are not absolutely convergent. So Fubini-Tonelli theorem does not apply here.
Is it sufficient that $int_a^infty f(x,y)dx, int_a^infty f(x,y)dy$ are uniformly convergent for $x,y in [0,infty)$? How is it proved if true?
real-analysis
asked Nov 30 '18 at 18:08
WoodWorkerWoodWorker
528311
$endgroup$
add a comment |
$begingroup$
I'm trying to understand general conditions that permit switching integrals as in
$$int_a^infty int_a^infty f(x,y) dx dy = int_a^infty int_a^infty f(x,y) dy dx $$
if $f$ is not nonnegative or nonpositive and the integrals $int_a^infty f(x,y)dx, int_a^infty f(x,y)dy$ are improper Riemann integrals that are not absolutely convergent. So Fubini-Tonelli theorem does not apply here.
Is it sufficient that $int_a^infty f(x,y)dx, int_a^infty f(x,y)dy$ are uniformly convergent for $x,y in [0,infty)$? How is it proved if true?
real-analysis
asked Nov 30 '18 at 18:08
WoodWorkerWoodWorker
528311
$endgroup$
add a comment |
$begingroup$
I'm trying to understand general conditions that permit switching integrals as in
$$int_a^infty int_a^infty f(x,y) dx dy = int_a^infty int_a^infty f(x,y) dy dx $$
if $f$ is not nonnegative or nonpositive and the integrals $int_a^infty f(x,y)dx, int_a^infty f(x,y)dy$ are improper Riemann integrals that are not absolutely convergent. So Fubini-Tonelli theorem does not apply here.
Is it sufficient that $int_a^infty f(x,y)dx, int_a^infty f(x,y)dy$ are uniformly convergent for $x,y in [0,infty)$? How is it proved if true?
real-analysis
asked Nov 30 '18 at 18:08
WoodWorkerWoodWorker
528311
$endgroup$
I'm trying to understand general conditions that permit switching integrals as in
$$int_a^infty int_a^infty f(x,y) dx dy = int_a^infty int_a^infty f(x,y) dy dx $$
if $f$ is not nonnegative or nonpositive and the integrals $int_a^infty f(x,y)dx, int_a^infty f(x,y)dy$ are improper Riemann integrals that are not absolutely convergent. So Fubini-Tonelli theorem does not apply here.
Is it sufficient that $int_a^infty f(x,y)dx, int_a^infty f(x,y)dy$ are uniformly convergent for $x,y in [0,infty)$? How is it proved if true?
real-analysis
real-analysis
asked Nov 30 '18 at 18:08
WoodWorkerWoodWorker
528311
asked Nov 30 '18 at 18:08
WoodWorkerWoodWorker
528311
asked Nov 30 '18 at 18:08
WoodWorkerWoodWorker
528311
asked Nov 30 '18 at 18:08
WoodWorkerWoodWorker
528311
asked Nov 30 '18 at 18:08
WoodWorkerWoodWorker
528311
528311
add a comment |
add a comment |
0
active
oldest
votes
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%2f3020423%2fswitching-improper-integrals-without-fubini%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3020423%2fswitching-improper-integrals-without-fubini%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
