Monotone Convergence Theorem and its Conditions
I am reading Folland's Real Analysis p. 51
Corollary:
If ${f_n} subset L^+$, $fin L^+$, and $f_n(x)$ increases to $f(x)$ for a.e. $x$, then $int f =lim int f_n $.
This is a bit relaxed from monotone convergence theorem since here we require only "a.e. x".
Then, the author gives two examples: $$chi_{(n,n+1)} rightarrow 0 text{and} n_{chi(0,1/n)}rightarrow 0 textbf{pointwise},$$
but $$int chi_{(n,n+1)} = int n_{chi(0,1/n)} = 1,$$
so obvious, both of them do not meet MCT (otherwise, both integration should be $0$). I cannot understand the explanation from the author.
Is this because of the difference between "pointwise" and "almost everywhere"?
real-analysis measure-theory lebesgue-integral
asked Dec 27 '18 at 7:53
sleeve chen
3,05941852
add a comment |
I am reading Folland's Real Analysis p. 51
Corollary:
If ${f_n} subset L^+$, $fin L^+$, and $f_n(x)$ increases to $f(x)$ for a.e. $x$, then $int f =lim int f_n $.
This is a bit relaxed from monotone convergence theorem since here we require only "a.e. x".
Then, the author gives two examples: $$chi_{(n,n+1)} rightarrow 0 text{and} n_{chi(0,1/n)}rightarrow 0 textbf{pointwise},$$
but $$int chi_{(n,n+1)} = int n_{chi(0,1/n)} = 1,$$
so obvious, both of them do not meet MCT (otherwise, both integration should be $0$). I cannot understand the explanation from the author.
Is this because of the difference between "pointwise" and "almost everywhere"?
real-analysis measure-theory lebesgue-integral
asked Dec 27 '18 at 7:53
sleeve chen
3,05941852
6
no... it's because of "increases"
– mathworker21
Dec 27 '18 at 8:03
add a comment |
I am reading Folland's Real Analysis p. 51
Corollary:
If ${f_n} subset L^+$, $fin L^+$, and $f_n(x)$ increases to $f(x)$ for a.e. $x$, then $int f =lim int f_n $.
This is a bit relaxed from monotone convergence theorem since here we require only "a.e. x".
Then, the author gives two examples: $$chi_{(n,n+1)} rightarrow 0 text{and} n_{chi(0,1/n)}rightarrow 0 textbf{pointwise},$$
but $$int chi_{(n,n+1)} = int n_{chi(0,1/n)} = 1,$$
so obvious, both of them do not meet MCT (otherwise, both integration should be $0$). I cannot understand the explanation from the author.
Is this because of the difference between "pointwise" and "almost everywhere"?
real-analysis measure-theory lebesgue-integral
asked Dec 27 '18 at 7:53
sleeve chen
3,05941852
I am reading Folland's Real Analysis p. 51
Corollary:
If ${f_n} subset L^+$, $fin L^+$, and $f_n(x)$ increases to $f(x)$ for a.e. $x$, then $int f =lim int f_n $.
This is a bit relaxed from monotone convergence theorem since here we require only "a.e. x".
Then, the author gives two examples: $$chi_{(n,n+1)} rightarrow 0 text{and} n_{chi(0,1/n)}rightarrow 0 textbf{pointwise},$$
but $$int chi_{(n,n+1)} = int n_{chi(0,1/n)} = 1,$$
so obvious, both of them do not meet MCT (otherwise, both integration should be $0$). I cannot understand the explanation from the author.
Is this because of the difference between "pointwise" and "almost everywhere"?
real-analysis measure-theory lebesgue-integral
real-analysis measure-theory lebesgue-integral
asked Dec 27 '18 at 7:53
sleeve chen
3,05941852
asked Dec 27 '18 at 7:53
sleeve chen
3,05941852
asked Dec 27 '18 at 7:53
sleeve chen
3,05941852
asked Dec 27 '18 at 7:53
sleeve chen
3,05941852
asked Dec 27 '18 at 7:53
sleeve chen
3,05941852
3,05941852
6
no... it's because of "increases"
– mathworker21
Dec 27 '18 at 8:03
add a comment |
6
no... it's because of "increases"
– mathworker21
Dec 27 '18 at 8:03
6
6
no... it's because of "increases"
– mathworker21
Dec 27 '18 at 8:03
no... it's because of "increases"
– mathworker21
Dec 27 '18 at 8:03
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%2f3053690%2fmonotone-convergence-theorem-and-its-conditions%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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3053690%2fmonotone-convergence-theorem-and-its-conditions%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
6
no... it's because of "increases"
– mathworker21
Dec 27 '18 at 8:03