Sequence of functions who are bounded from below, change of limit and integral
$begingroup$
I have a question about changing limit and integral. I know of the monotone convergence theorem, so if my sequence is greater zero and increasing, I can change integral and limit. My question now is, what if my sequence is only bounded from below, by a possible negative number, and increasing. Can I change integral and limit is this case? I am pretty sure I can't. Does something change if I have a probability space? Thanks in advance!
real-analysis probability measure-theory
asked Dec 14 '18 at 11:38
Testname420Testname420
31
$endgroup$
add a comment |
$begingroup$
I have a question about changing limit and integral. I know of the monotone convergence theorem, so if my sequence is greater zero and increasing, I can change integral and limit. My question now is, what if my sequence is only bounded from below, by a possible negative number, and increasing. Can I change integral and limit is this case? I am pretty sure I can't. Does something change if I have a probability space? Thanks in advance!
real-analysis probability measure-theory
asked Dec 14 '18 at 11:38
Testname420Testname420
31
$endgroup$
add a comment |
$begingroup$
I have a question about changing limit and integral. I know of the monotone convergence theorem, so if my sequence is greater zero and increasing, I can change integral and limit. My question now is, what if my sequence is only bounded from below, by a possible negative number, and increasing. Can I change integral and limit is this case? I am pretty sure I can't. Does something change if I have a probability space? Thanks in advance!
real-analysis probability measure-theory
asked Dec 14 '18 at 11:38
Testname420Testname420
31
$endgroup$
I have a question about changing limit and integral. I know of the monotone convergence theorem, so if my sequence is greater zero and increasing, I can change integral and limit. My question now is, what if my sequence is only bounded from below, by a possible negative number, and increasing. Can I change integral and limit is this case? I am pretty sure I can't. Does something change if I have a probability space? Thanks in advance!
real-analysis probability measure-theory
real-analysis probability measure-theory
asked Dec 14 '18 at 11:38
Testname420Testname420
31
asked Dec 14 '18 at 11:38
Testname420Testname420
31
asked Dec 14 '18 at 11:38
Testname420Testname420
31
asked Dec 14 '18 at 11:38
Testname420Testname420
31
asked Dec 14 '18 at 11:38
Testname420Testname420
31
31
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$-I_{(n, infty)}$ is a counter example for the general case. In the case of finite measure $f_n$ increasing and $geq -a$ implies $f_n+a $ is non-negative and increasing so $lim int (f_n+a) to int (f+a)$ and we can subtract $a mu (X)$ from both sides to get $lim int f_n to int f$
answered Dec 14 '18 at 11:43
Kavi Rama MurthyKavi Rama Murthy
59.4k42161
$endgroup$
$begingroup$
Alright makes a lot of sense thank you
$endgroup$
– Testname420
Dec 14 '18 at 11:53
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%2f3039256%2fsequence-of-functions-who-are-bounded-from-below-change-of-limit-and-integral%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
$begingroup$
$-I_{(n, infty)}$ is a counter example for the general case. In the case of finite measure $f_n$ increasing and $geq -a$ implies $f_n+a $ is non-negative and increasing so $lim int (f_n+a) to int (f+a)$ and we can subtract $a mu (X)$ from both sides to get $lim int f_n to int f$
answered Dec 14 '18 at 11:43
Kavi Rama MurthyKavi Rama Murthy
59.4k42161
$endgroup$
$begingroup$
Alright makes a lot of sense thank you
$endgroup$
– Testname420
Dec 14 '18 at 11:53
add a comment |
$begingroup$
$-I_{(n, infty)}$ is a counter example for the general case. In the case of finite measure $f_n$ increasing and $geq -a$ implies $f_n+a $ is non-negative and increasing so $lim int (f_n+a) to int (f+a)$ and we can subtract $a mu (X)$ from both sides to get $lim int f_n to int f$
answered Dec 14 '18 at 11:43
Kavi Rama MurthyKavi Rama Murthy
59.4k42161
$endgroup$
$begingroup$
Alright makes a lot of sense thank you
$endgroup$
– Testname420
Dec 14 '18 at 11:53
add a comment |
$begingroup$
$-I_{(n, infty)}$ is a counter example for the general case. In the case of finite measure $f_n$ increasing and $geq -a$ implies $f_n+a $ is non-negative and increasing so $lim int (f_n+a) to int (f+a)$ and we can subtract $a mu (X)$ from both sides to get $lim int f_n to int f$
answered Dec 14 '18 at 11:43
Kavi Rama MurthyKavi Rama Murthy
59.4k42161
$endgroup$
$-I_{(n, infty)}$ is a counter example for the general case. In the case of finite measure $f_n$ increasing and $geq -a$ implies $f_n+a $ is non-negative and increasing so $lim int (f_n+a) to int (f+a)$ and we can subtract $a mu (X)$ from both sides to get $lim int f_n to int f$
answered Dec 14 '18 at 11:43
Kavi Rama MurthyKavi Rama Murthy
59.4k42161
answered Dec 14 '18 at 11:43
Kavi Rama MurthyKavi Rama Murthy
59.4k42161
answered Dec 14 '18 at 11:43
Kavi Rama MurthyKavi Rama Murthy
59.4k42161
answered Dec 14 '18 at 11:43
Kavi Rama MurthyKavi Rama Murthy
59.4k42161
59.4k42161
$begingroup$
Alright makes a lot of sense thank you
$endgroup$
– Testname420
Dec 14 '18 at 11:53
add a comment |
$begingroup$
Alright makes a lot of sense thank you
$endgroup$
– Testname420
Dec 14 '18 at 11:53
$begingroup$
Alright makes a lot of sense thank you
$endgroup$
– Testname420
Dec 14 '18 at 11:53
$begingroup$
Alright makes a lot of sense thank you
$endgroup$
– Testname420
Dec 14 '18 at 11:53
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%2f3039256%2fsequence-of-functions-who-are-bounded-from-below-change-of-limit-and-integral%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
