With what extra condition interchange limit and integration allowed?
$begingroup$
$f_n$ is sequence of continuous function which converges pointwise to continuous function.
with what condition we can interchange limit and integration?
I know that if sequence converges uniformly then we can.But I wanted some what weaker condition .
Any Help will be appreciated.
real-analysis sequences-and-series pointwise-convergence
asked Jan 18 at 13:38
MathLoverMathLover
58612
$endgroup$
add a comment |
$begingroup$
$f_n$ is sequence of continuous function which converges pointwise to continuous function.
with what condition we can interchange limit and integration?
I know that if sequence converges uniformly then we can.But I wanted some what weaker condition .
Any Help will be appreciated.
real-analysis sequences-and-series pointwise-convergence
asked Jan 18 at 13:38
MathLoverMathLover
58612
$endgroup$
add a comment |
$begingroup$
$f_n$ is sequence of continuous function which converges pointwise to continuous function.
with what condition we can interchange limit and integration?
I know that if sequence converges uniformly then we can.But I wanted some what weaker condition .
Any Help will be appreciated.
real-analysis sequences-and-series pointwise-convergence
asked Jan 18 at 13:38
MathLoverMathLover
58612
$endgroup$
$f_n$ is sequence of continuous function which converges pointwise to continuous function.
with what condition we can interchange limit and integration?
I know that if sequence converges uniformly then we can.But I wanted some what weaker condition .
Any Help will be appreciated.
real-analysis sequences-and-series pointwise-convergence
real-analysis sequences-and-series pointwise-convergence
asked Jan 18 at 13:38
MathLoverMathLover
58612
asked Jan 18 at 13:38
MathLoverMathLover
58612
asked Jan 18 at 13:38
MathLoverMathLover
58612
asked Jan 18 at 13:38
MathLoverMathLover
58612
asked Jan 18 at 13:38
MathLoverMathLover
58612
58612
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You might be looking for the dominated convergence theorem, which states that the interchange is permitted if there is an integrable function $g$ such that $|f_n(x)| leq g(x)$ for all $n in mathbb{N}$.
answered Jan 18 at 13:54
KlausKlaus
2,955214
$endgroup$
add a comment |
Your Answer
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%2f3078257%2fwith-what-extra-condition-interchange-limit-and-integration-allowed%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
$begingroup$
You might be looking for the dominated convergence theorem, which states that the interchange is permitted if there is an integrable function $g$ such that $|f_n(x)| leq g(x)$ for all $n in mathbb{N}$.
answered Jan 18 at 13:54
KlausKlaus
2,955214
$endgroup$
add a comment |
$begingroup$
You might be looking for the dominated convergence theorem, which states that the interchange is permitted if there is an integrable function $g$ such that $|f_n(x)| leq g(x)$ for all $n in mathbb{N}$.
answered Jan 18 at 13:54
KlausKlaus
2,955214
$endgroup$
add a comment |
$begingroup$
You might be looking for the dominated convergence theorem, which states that the interchange is permitted if there is an integrable function $g$ such that $|f_n(x)| leq g(x)$ for all $n in mathbb{N}$.
answered Jan 18 at 13:54
KlausKlaus
2,955214
$endgroup$
You might be looking for the dominated convergence theorem, which states that the interchange is permitted if there is an integrable function $g$ such that $|f_n(x)| leq g(x)$ for all $n in mathbb{N}$.
answered Jan 18 at 13:54
KlausKlaus
2,955214
answered Jan 18 at 13:54
KlausKlaus
2,955214
answered Jan 18 at 13:54
KlausKlaus
2,955214
answered Jan 18 at 13:54
KlausKlaus
2,955214
2,955214
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%2f3078257%2fwith-what-extra-condition-interchange-limit-and-integration-allowed%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
