$lambda$-almost-everywhere convergence implied by lim$lambda (${$x in E : | f_n(x) - f(x) | > epsilon$}$)...












1












$begingroup$


Let $E subset mathbb{R}$ be $lambda$-measurable and let $f_n,f: E -> mathbb{R}$ $lambda$-integrable, so that for all $epsilon > 0$



$lambda(${$xin E: |f_n(x)-f(x)| > epsilon$}$)->0$ as $n -> infty$



Proof that $f_n$ converges to $f$ $lambda$-almost-everywhere.



My attempt:



Assume $f_n$ does not converge almost everywhere to $f$ then there exists an $ epsilon$ for which for all $N in mathbb{N}$ there's a $n>=N$ with $|f_n(x)-f(x)|>epsilon$ for all $x in A$ with $A$ being a set that's measure isn't zero.



Then x is in infinitely many $H_n^epsilon:=${$x in E : |f_n(x)-f(x)|>epsilon$} and thus $sum_{n=1}^inftylambda(H_n^epsilon)=infty$.



On the other hand, since $f_n,f$ are integrable we get:



$infty>int_E|f_n(x)-f(x)|dlambda>int_{H_n^epsilon}|f_n(x)-f(x)|dlambda=sum_{n}int_{H_n^epsilon}|f_n(x)-f(x)|dlambda>sum_{n}epsilon lambda(H_n^epsilon)=infty$



What is a contradiction. And the assumption must be wrong.



I guess this is wrong as I didn't even use $lambda(${$xin E: |f_n(x)-f(x)| > epsilon$}$)->0$ as $n -> infty$. However, I do not find my mistake nor have I an idea on how to approach this problem.



I would really appreciate any help.










share|cite|improve this question









$endgroup$












  • $begingroup$
    I didn't read your attempt, but maybe you want to read en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma. This is the standard tool to prove it
    $endgroup$
    – Federico
    Dec 4 '18 at 19:33












  • $begingroup$
    I think your statement of what it means to not converge almost everywhere is not quite right. It seems like you're claiming: There exists $epsilon > 0$ and a set of positive measure $A$ such that, for any $N in mathbb{N}$ there is $n > N$ with $|f_n(x) - f(x)| > epsilon$ for all $x in A$. Rather I think it should be the following: There exists a set $A$ of positive measure, such that for all $x in A$ there exists an $epsilon > 0$ such that, for any $N in mathbb{N}$ there exists $n > N$ with $|f_n(x) - f(x)| geq epsilon$.
    $endgroup$
    – Sean Haight
    Dec 4 '18 at 19:53












  • $begingroup$
    @SeanHaight you're right.
    $endgroup$
    – Kekks
    Dec 4 '18 at 20:06










  • $begingroup$
    @Federico so i use the integrability of $f_n,f$ to shot that $sum_n lambda(H_n^{epsilon}) < infty$ so i can use Borel-Cantelli which implies the convergence a.e.?
    $endgroup$
    – Kekks
    Dec 4 '18 at 20:06












  • $begingroup$
    You write the set of non-convergence as an intersection of a union, like in the statement of Borel-Cantelli lemma, then use it to show that it has measure $0$.
    $endgroup$
    – Federico
    Dec 4 '18 at 20:09
















1












$begingroup$


Let $E subset mathbb{R}$ be $lambda$-measurable and let $f_n,f: E -> mathbb{R}$ $lambda$-integrable, so that for all $epsilon > 0$



$lambda(${$xin E: |f_n(x)-f(x)| > epsilon$}$)->0$ as $n -> infty$



Proof that $f_n$ converges to $f$ $lambda$-almost-everywhere.



My attempt:



Assume $f_n$ does not converge almost everywhere to $f$ then there exists an $ epsilon$ for which for all $N in mathbb{N}$ there's a $n>=N$ with $|f_n(x)-f(x)|>epsilon$ for all $x in A$ with $A$ being a set that's measure isn't zero.



Then x is in infinitely many $H_n^epsilon:=${$x in E : |f_n(x)-f(x)|>epsilon$} and thus $sum_{n=1}^inftylambda(H_n^epsilon)=infty$.



On the other hand, since $f_n,f$ are integrable we get:



$infty>int_E|f_n(x)-f(x)|dlambda>int_{H_n^epsilon}|f_n(x)-f(x)|dlambda=sum_{n}int_{H_n^epsilon}|f_n(x)-f(x)|dlambda>sum_{n}epsilon lambda(H_n^epsilon)=infty$



What is a contradiction. And the assumption must be wrong.



I guess this is wrong as I didn't even use $lambda(${$xin E: |f_n(x)-f(x)| > epsilon$}$)->0$ as $n -> infty$. However, I do not find my mistake nor have I an idea on how to approach this problem.



I would really appreciate any help.










share|cite|improve this question









$endgroup$












  • $begingroup$
    I didn't read your attempt, but maybe you want to read en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma. This is the standard tool to prove it
    $endgroup$
    – Federico
    Dec 4 '18 at 19:33












  • $begingroup$
    I think your statement of what it means to not converge almost everywhere is not quite right. It seems like you're claiming: There exists $epsilon > 0$ and a set of positive measure $A$ such that, for any $N in mathbb{N}$ there is $n > N$ with $|f_n(x) - f(x)| > epsilon$ for all $x in A$. Rather I think it should be the following: There exists a set $A$ of positive measure, such that for all $x in A$ there exists an $epsilon > 0$ such that, for any $N in mathbb{N}$ there exists $n > N$ with $|f_n(x) - f(x)| geq epsilon$.
    $endgroup$
    – Sean Haight
    Dec 4 '18 at 19:53












  • $begingroup$
    @SeanHaight you're right.
    $endgroup$
    – Kekks
    Dec 4 '18 at 20:06










  • $begingroup$
    @Federico so i use the integrability of $f_n,f$ to shot that $sum_n lambda(H_n^{epsilon}) < infty$ so i can use Borel-Cantelli which implies the convergence a.e.?
    $endgroup$
    – Kekks
    Dec 4 '18 at 20:06












  • $begingroup$
    You write the set of non-convergence as an intersection of a union, like in the statement of Borel-Cantelli lemma, then use it to show that it has measure $0$.
    $endgroup$
    – Federico
    Dec 4 '18 at 20:09














1












1








1





$begingroup$


Let $E subset mathbb{R}$ be $lambda$-measurable and let $f_n,f: E -> mathbb{R}$ $lambda$-integrable, so that for all $epsilon > 0$



$lambda(${$xin E: |f_n(x)-f(x)| > epsilon$}$)->0$ as $n -> infty$



Proof that $f_n$ converges to $f$ $lambda$-almost-everywhere.



My attempt:



Assume $f_n$ does not converge almost everywhere to $f$ then there exists an $ epsilon$ for which for all $N in mathbb{N}$ there's a $n>=N$ with $|f_n(x)-f(x)|>epsilon$ for all $x in A$ with $A$ being a set that's measure isn't zero.



Then x is in infinitely many $H_n^epsilon:=${$x in E : |f_n(x)-f(x)|>epsilon$} and thus $sum_{n=1}^inftylambda(H_n^epsilon)=infty$.



On the other hand, since $f_n,f$ are integrable we get:



$infty>int_E|f_n(x)-f(x)|dlambda>int_{H_n^epsilon}|f_n(x)-f(x)|dlambda=sum_{n}int_{H_n^epsilon}|f_n(x)-f(x)|dlambda>sum_{n}epsilon lambda(H_n^epsilon)=infty$



What is a contradiction. And the assumption must be wrong.



I guess this is wrong as I didn't even use $lambda(${$xin E: |f_n(x)-f(x)| > epsilon$}$)->0$ as $n -> infty$. However, I do not find my mistake nor have I an idea on how to approach this problem.



I would really appreciate any help.










share|cite|improve this question









$endgroup$




Let $E subset mathbb{R}$ be $lambda$-measurable and let $f_n,f: E -> mathbb{R}$ $lambda$-integrable, so that for all $epsilon > 0$



$lambda(${$xin E: |f_n(x)-f(x)| > epsilon$}$)->0$ as $n -> infty$



Proof that $f_n$ converges to $f$ $lambda$-almost-everywhere.



My attempt:



Assume $f_n$ does not converge almost everywhere to $f$ then there exists an $ epsilon$ for which for all $N in mathbb{N}$ there's a $n>=N$ with $|f_n(x)-f(x)|>epsilon$ for all $x in A$ with $A$ being a set that's measure isn't zero.



Then x is in infinitely many $H_n^epsilon:=${$x in E : |f_n(x)-f(x)|>epsilon$} and thus $sum_{n=1}^inftylambda(H_n^epsilon)=infty$.



On the other hand, since $f_n,f$ are integrable we get:



$infty>int_E|f_n(x)-f(x)|dlambda>int_{H_n^epsilon}|f_n(x)-f(x)|dlambda=sum_{n}int_{H_n^epsilon}|f_n(x)-f(x)|dlambda>sum_{n}epsilon lambda(H_n^epsilon)=infty$



What is a contradiction. And the assumption must be wrong.



I guess this is wrong as I didn't even use $lambda(${$xin E: |f_n(x)-f(x)| > epsilon$}$)->0$ as $n -> infty$. However, I do not find my mistake nor have I an idea on how to approach this problem.



I would really appreciate any help.







measure-theory lebesgue-measure almost-everywhere






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 4 '18 at 19:26









KekksKekks

418




418












  • $begingroup$
    I didn't read your attempt, but maybe you want to read en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma. This is the standard tool to prove it
    $endgroup$
    – Federico
    Dec 4 '18 at 19:33












  • $begingroup$
    I think your statement of what it means to not converge almost everywhere is not quite right. It seems like you're claiming: There exists $epsilon > 0$ and a set of positive measure $A$ such that, for any $N in mathbb{N}$ there is $n > N$ with $|f_n(x) - f(x)| > epsilon$ for all $x in A$. Rather I think it should be the following: There exists a set $A$ of positive measure, such that for all $x in A$ there exists an $epsilon > 0$ such that, for any $N in mathbb{N}$ there exists $n > N$ with $|f_n(x) - f(x)| geq epsilon$.
    $endgroup$
    – Sean Haight
    Dec 4 '18 at 19:53












  • $begingroup$
    @SeanHaight you're right.
    $endgroup$
    – Kekks
    Dec 4 '18 at 20:06










  • $begingroup$
    @Federico so i use the integrability of $f_n,f$ to shot that $sum_n lambda(H_n^{epsilon}) < infty$ so i can use Borel-Cantelli which implies the convergence a.e.?
    $endgroup$
    – Kekks
    Dec 4 '18 at 20:06












  • $begingroup$
    You write the set of non-convergence as an intersection of a union, like in the statement of Borel-Cantelli lemma, then use it to show that it has measure $0$.
    $endgroup$
    – Federico
    Dec 4 '18 at 20:09


















  • $begingroup$
    I didn't read your attempt, but maybe you want to read en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma. This is the standard tool to prove it
    $endgroup$
    – Federico
    Dec 4 '18 at 19:33












  • $begingroup$
    I think your statement of what it means to not converge almost everywhere is not quite right. It seems like you're claiming: There exists $epsilon > 0$ and a set of positive measure $A$ such that, for any $N in mathbb{N}$ there is $n > N$ with $|f_n(x) - f(x)| > epsilon$ for all $x in A$. Rather I think it should be the following: There exists a set $A$ of positive measure, such that for all $x in A$ there exists an $epsilon > 0$ such that, for any $N in mathbb{N}$ there exists $n > N$ with $|f_n(x) - f(x)| geq epsilon$.
    $endgroup$
    – Sean Haight
    Dec 4 '18 at 19:53












  • $begingroup$
    @SeanHaight you're right.
    $endgroup$
    – Kekks
    Dec 4 '18 at 20:06










  • $begingroup$
    @Federico so i use the integrability of $f_n,f$ to shot that $sum_n lambda(H_n^{epsilon}) < infty$ so i can use Borel-Cantelli which implies the convergence a.e.?
    $endgroup$
    – Kekks
    Dec 4 '18 at 20:06












  • $begingroup$
    You write the set of non-convergence as an intersection of a union, like in the statement of Borel-Cantelli lemma, then use it to show that it has measure $0$.
    $endgroup$
    – Federico
    Dec 4 '18 at 20:09
















$begingroup$
I didn't read your attempt, but maybe you want to read en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma. This is the standard tool to prove it
$endgroup$
– Federico
Dec 4 '18 at 19:33






$begingroup$
I didn't read your attempt, but maybe you want to read en.wikipedia.org/wiki/Borel%E2%80%93Cantelli_lemma. This is the standard tool to prove it
$endgroup$
– Federico
Dec 4 '18 at 19:33














$begingroup$
I think your statement of what it means to not converge almost everywhere is not quite right. It seems like you're claiming: There exists $epsilon > 0$ and a set of positive measure $A$ such that, for any $N in mathbb{N}$ there is $n > N$ with $|f_n(x) - f(x)| > epsilon$ for all $x in A$. Rather I think it should be the following: There exists a set $A$ of positive measure, such that for all $x in A$ there exists an $epsilon > 0$ such that, for any $N in mathbb{N}$ there exists $n > N$ with $|f_n(x) - f(x)| geq epsilon$.
$endgroup$
– Sean Haight
Dec 4 '18 at 19:53






$begingroup$
I think your statement of what it means to not converge almost everywhere is not quite right. It seems like you're claiming: There exists $epsilon > 0$ and a set of positive measure $A$ such that, for any $N in mathbb{N}$ there is $n > N$ with $|f_n(x) - f(x)| > epsilon$ for all $x in A$. Rather I think it should be the following: There exists a set $A$ of positive measure, such that for all $x in A$ there exists an $epsilon > 0$ such that, for any $N in mathbb{N}$ there exists $n > N$ with $|f_n(x) - f(x)| geq epsilon$.
$endgroup$
– Sean Haight
Dec 4 '18 at 19:53














$begingroup$
@SeanHaight you're right.
$endgroup$
– Kekks
Dec 4 '18 at 20:06




$begingroup$
@SeanHaight you're right.
$endgroup$
– Kekks
Dec 4 '18 at 20:06












$begingroup$
@Federico so i use the integrability of $f_n,f$ to shot that $sum_n lambda(H_n^{epsilon}) < infty$ so i can use Borel-Cantelli which implies the convergence a.e.?
$endgroup$
– Kekks
Dec 4 '18 at 20:06






$begingroup$
@Federico so i use the integrability of $f_n,f$ to shot that $sum_n lambda(H_n^{epsilon}) < infty$ so i can use Borel-Cantelli which implies the convergence a.e.?
$endgroup$
– Kekks
Dec 4 '18 at 20:06














$begingroup$
You write the set of non-convergence as an intersection of a union, like in the statement of Borel-Cantelli lemma, then use it to show that it has measure $0$.
$endgroup$
– Federico
Dec 4 '18 at 20:09




$begingroup$
You write the set of non-convergence as an intersection of a union, like in the statement of Borel-Cantelli lemma, then use it to show that it has measure $0$.
$endgroup$
– Federico
Dec 4 '18 at 20:09










1 Answer
1






active

oldest

votes


















0












$begingroup$

Fix $k$ and pick $n_k$ such that $lambda E_k le {1 over 2^k}$ where $E_k ={ x | |f_{n_k}(x)-f(x)| > { 1over k} }$.



This is the Borel Cantelli lemma:
Note that $int sum_k 1_{E_k} = sum_k int 1_{E_k} = sum_{k} lambda E_k < infty$
and so $sum_k 1_{E_k}$ is finite ae. Hence for ae. $x$, $x$ is in at most a finite number of $E_k$ and so $f_{n_k}(x) to f(x)$.



Addendum:



The convergence cannot be everywhere. Here is an example that is easier to describe than to write an explicit expression. Choose the space $[0,1]$, let $f = 0$
and define $f_n$ as follows: $f_1 = 1_{[0,{1 over 2}]}, f_2 = 1_{[{1 over 2},1]}$, $f_3 = 1_{[0,{1 over 4}]}$, $f_4 = 1_{[{1 over 4}, {1 over 2}]}$, $f_5 = 1_{[{1 over 2}, {3 over 4}]}$, etc. It should be clear that $lambda { x | |f_n(x)| > epsilon} to 0$, but for every $x in [0,1)$, we have $f_n(x) = 1$ infinitely often and $f_n(x) = 0$ and infinitely often.



However, if we pick the subsequence corresponding to the functions $1_{[0,{1 over 2^n}]}$ then we see that this subsequence converges pointwise to $0$ for $x in (0,1]$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    This only proves that a subsequence converges, isnt it?
    $endgroup$
    – Kekks
    Dec 4 '18 at 21:24










  • $begingroup$
    Correct. ${}{}{}$
    $endgroup$
    – copper.hat
    Dec 4 '18 at 21:25











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3026021%2flambda-almost-everywhere-convergence-implied-by-lim-lambda-x-in-e-f%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









0












$begingroup$

Fix $k$ and pick $n_k$ such that $lambda E_k le {1 over 2^k}$ where $E_k ={ x | |f_{n_k}(x)-f(x)| > { 1over k} }$.



This is the Borel Cantelli lemma:
Note that $int sum_k 1_{E_k} = sum_k int 1_{E_k} = sum_{k} lambda E_k < infty$
and so $sum_k 1_{E_k}$ is finite ae. Hence for ae. $x$, $x$ is in at most a finite number of $E_k$ and so $f_{n_k}(x) to f(x)$.



Addendum:



The convergence cannot be everywhere. Here is an example that is easier to describe than to write an explicit expression. Choose the space $[0,1]$, let $f = 0$
and define $f_n$ as follows: $f_1 = 1_{[0,{1 over 2}]}, f_2 = 1_{[{1 over 2},1]}$, $f_3 = 1_{[0,{1 over 4}]}$, $f_4 = 1_{[{1 over 4}, {1 over 2}]}$, $f_5 = 1_{[{1 over 2}, {3 over 4}]}$, etc. It should be clear that $lambda { x | |f_n(x)| > epsilon} to 0$, but for every $x in [0,1)$, we have $f_n(x) = 1$ infinitely often and $f_n(x) = 0$ and infinitely often.



However, if we pick the subsequence corresponding to the functions $1_{[0,{1 over 2^n}]}$ then we see that this subsequence converges pointwise to $0$ for $x in (0,1]$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    This only proves that a subsequence converges, isnt it?
    $endgroup$
    – Kekks
    Dec 4 '18 at 21:24










  • $begingroup$
    Correct. ${}{}{}$
    $endgroup$
    – copper.hat
    Dec 4 '18 at 21:25
















0












$begingroup$

Fix $k$ and pick $n_k$ such that $lambda E_k le {1 over 2^k}$ where $E_k ={ x | |f_{n_k}(x)-f(x)| > { 1over k} }$.



This is the Borel Cantelli lemma:
Note that $int sum_k 1_{E_k} = sum_k int 1_{E_k} = sum_{k} lambda E_k < infty$
and so $sum_k 1_{E_k}$ is finite ae. Hence for ae. $x$, $x$ is in at most a finite number of $E_k$ and so $f_{n_k}(x) to f(x)$.



Addendum:



The convergence cannot be everywhere. Here is an example that is easier to describe than to write an explicit expression. Choose the space $[0,1]$, let $f = 0$
and define $f_n$ as follows: $f_1 = 1_{[0,{1 over 2}]}, f_2 = 1_{[{1 over 2},1]}$, $f_3 = 1_{[0,{1 over 4}]}$, $f_4 = 1_{[{1 over 4}, {1 over 2}]}$, $f_5 = 1_{[{1 over 2}, {3 over 4}]}$, etc. It should be clear that $lambda { x | |f_n(x)| > epsilon} to 0$, but for every $x in [0,1)$, we have $f_n(x) = 1$ infinitely often and $f_n(x) = 0$ and infinitely often.



However, if we pick the subsequence corresponding to the functions $1_{[0,{1 over 2^n}]}$ then we see that this subsequence converges pointwise to $0$ for $x in (0,1]$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    This only proves that a subsequence converges, isnt it?
    $endgroup$
    – Kekks
    Dec 4 '18 at 21:24










  • $begingroup$
    Correct. ${}{}{}$
    $endgroup$
    – copper.hat
    Dec 4 '18 at 21:25














0












0








0





$begingroup$

Fix $k$ and pick $n_k$ such that $lambda E_k le {1 over 2^k}$ where $E_k ={ x | |f_{n_k}(x)-f(x)| > { 1over k} }$.



This is the Borel Cantelli lemma:
Note that $int sum_k 1_{E_k} = sum_k int 1_{E_k} = sum_{k} lambda E_k < infty$
and so $sum_k 1_{E_k}$ is finite ae. Hence for ae. $x$, $x$ is in at most a finite number of $E_k$ and so $f_{n_k}(x) to f(x)$.



Addendum:



The convergence cannot be everywhere. Here is an example that is easier to describe than to write an explicit expression. Choose the space $[0,1]$, let $f = 0$
and define $f_n$ as follows: $f_1 = 1_{[0,{1 over 2}]}, f_2 = 1_{[{1 over 2},1]}$, $f_3 = 1_{[0,{1 over 4}]}$, $f_4 = 1_{[{1 over 4}, {1 over 2}]}$, $f_5 = 1_{[{1 over 2}, {3 over 4}]}$, etc. It should be clear that $lambda { x | |f_n(x)| > epsilon} to 0$, but for every $x in [0,1)$, we have $f_n(x) = 1$ infinitely often and $f_n(x) = 0$ and infinitely often.



However, if we pick the subsequence corresponding to the functions $1_{[0,{1 over 2^n}]}$ then we see that this subsequence converges pointwise to $0$ for $x in (0,1]$.






share|cite|improve this answer











$endgroup$



Fix $k$ and pick $n_k$ such that $lambda E_k le {1 over 2^k}$ where $E_k ={ x | |f_{n_k}(x)-f(x)| > { 1over k} }$.



This is the Borel Cantelli lemma:
Note that $int sum_k 1_{E_k} = sum_k int 1_{E_k} = sum_{k} lambda E_k < infty$
and so $sum_k 1_{E_k}$ is finite ae. Hence for ae. $x$, $x$ is in at most a finite number of $E_k$ and so $f_{n_k}(x) to f(x)$.



Addendum:



The convergence cannot be everywhere. Here is an example that is easier to describe than to write an explicit expression. Choose the space $[0,1]$, let $f = 0$
and define $f_n$ as follows: $f_1 = 1_{[0,{1 over 2}]}, f_2 = 1_{[{1 over 2},1]}$, $f_3 = 1_{[0,{1 over 4}]}$, $f_4 = 1_{[{1 over 4}, {1 over 2}]}$, $f_5 = 1_{[{1 over 2}, {3 over 4}]}$, etc. It should be clear that $lambda { x | |f_n(x)| > epsilon} to 0$, but for every $x in [0,1)$, we have $f_n(x) = 1$ infinitely often and $f_n(x) = 0$ and infinitely often.



However, if we pick the subsequence corresponding to the functions $1_{[0,{1 over 2^n}]}$ then we see that this subsequence converges pointwise to $0$ for $x in (0,1]$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 4 '18 at 21:45

























answered Dec 4 '18 at 21:16









copper.hatcopper.hat

126k559160




126k559160












  • $begingroup$
    This only proves that a subsequence converges, isnt it?
    $endgroup$
    – Kekks
    Dec 4 '18 at 21:24










  • $begingroup$
    Correct. ${}{}{}$
    $endgroup$
    – copper.hat
    Dec 4 '18 at 21:25


















  • $begingroup$
    This only proves that a subsequence converges, isnt it?
    $endgroup$
    – Kekks
    Dec 4 '18 at 21:24










  • $begingroup$
    Correct. ${}{}{}$
    $endgroup$
    – copper.hat
    Dec 4 '18 at 21:25
















$begingroup$
This only proves that a subsequence converges, isnt it?
$endgroup$
– Kekks
Dec 4 '18 at 21:24




$begingroup$
This only proves that a subsequence converges, isnt it?
$endgroup$
– Kekks
Dec 4 '18 at 21:24












$begingroup$
Correct. ${}{}{}$
$endgroup$
– copper.hat
Dec 4 '18 at 21:25




$begingroup$
Correct. ${}{}{}$
$endgroup$
– copper.hat
Dec 4 '18 at 21:25


















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3026021%2flambda-almost-everywhere-convergence-implied-by-lim-lambda-x-in-e-f%23new-answer', 'question_page');
}
);

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







Popular posts from this blog

Probability when a professor distributes a quiz and homework assignment to a class of n students.

Aardman Animations

Are they similar matrix