Sequence of functions on $mathcal{L}^1([0,1])$ with $lim_{nrightarrow infty}||f_n||_1=0$ but $sup{f_n(x):...












1












$begingroup$


Problem Statement.



Good evening. As the title suggests, I am having a hard time with the following exercise:




Prove that there exists a sequence $f_1,f_2,ldots$ of functions on
$mathcal{L}^1([0,1])$ such that $lim_{nrightarrow
infty}||f_n||_1=0$ but $$sup{f_n(x): ninmathbf{Z}^+}=infty$$
for every $xin [0,1].$





Notation.



Here $mathcal{L}^1([0,1])$ means $mathcal{L}^1(lambda_{[0,1]})$ where $lambda_{[0,1]}$ represents Lebesgue measure restricted to the Borel subsets of $mathbf{R}$ that are contained in $[0,1]$.




Questions.



Ok, so my first question is how do I interpret $lim_{nrightarrow
infty}||f_n||_1=0$? I know that by definition $$||f||_1=int |f|,dmu$$ so I think this statement means $$lim_{nrightarrow infty}||f_n||=lim_{nrightarrowinfty}int|f_n|,dmu=0.$$ But I haven't been able to get anywhere with this, possibly because of my confusion with the other part of the question, which is even finding a sequence of functions that have the property $sup{f_n(x): ninmathbf{Z}^+}=infty$. I initially tried playing around with functions like $$f_n(x)=frac{1}{sqrt{nx}}$$ but this is undefined at $x=0$. So to summarize, my two questions are:




  1. Understanding the limit notation I pointed out.

  2. Finding a sequence of functions with an undefined supremum at every $x$.


Any suggestions towards either of these points of confusion would be wonderful.



Thank you for your time!






real-analysis sequences-and-series measure-theory lp-spaces supremum-and-infimum







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Try a sequence of functions where the height of the functions grows, but the size of the interval shrinks quickly. This will bound the area, but also make the height(and thus the sup) grow.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:21










  • $begingroup$
    @rubikscube09 Hmmm, ok. Would something like $f_n(x)=e^{nx}$ work?
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:24










  • $begingroup$
    @rubikscube09 I think the integral on that would grow instead of decay though hmmm... I'll keep trying! Thanks for the suggestion!
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:26








  • 1




    $begingroup$
    Do you know about indicator functions? They might be useful for something like this.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:28






  • 2




    $begingroup$
    Try to see how you could represent thin spikes that keep getting thinner and taller with indicator functions.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:34
















1












$begingroup$


Problem Statement.



Good evening. As the title suggests, I am having a hard time with the following exercise:




Prove that there exists a sequence $f_1,f_2,ldots$ of functions on
$mathcal{L}^1([0,1])$ such that $lim_{nrightarrow
infty}||f_n||_1=0$ but $$sup{f_n(x): ninmathbf{Z}^+}=infty$$
for every $xin [0,1].$





Notation.



Here $mathcal{L}^1([0,1])$ means $mathcal{L}^1(lambda_{[0,1]})$ where $lambda_{[0,1]}$ represents Lebesgue measure restricted to the Borel subsets of $mathbf{R}$ that are contained in $[0,1]$.




Questions.



Ok, so my first question is how do I interpret $lim_{nrightarrow
infty}||f_n||_1=0$? I know that by definition $$||f||_1=int |f|,dmu$$ so I think this statement means $$lim_{nrightarrow infty}||f_n||=lim_{nrightarrowinfty}int|f_n|,dmu=0.$$ But I haven't been able to get anywhere with this, possibly because of my confusion with the other part of the question, which is even finding a sequence of functions that have the property $sup{f_n(x): ninmathbf{Z}^+}=infty$. I initially tried playing around with functions like $$f_n(x)=frac{1}{sqrt{nx}}$$ but this is undefined at $x=0$. So to summarize, my two questions are:




  1. Understanding the limit notation I pointed out.

  2. Finding a sequence of functions with an undefined supremum at every $x$.


Any suggestions towards either of these points of confusion would be wonderful.



Thank you for your time!






real-analysis sequences-and-series measure-theory lp-spaces supremum-and-infimum







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Try a sequence of functions where the height of the functions grows, but the size of the interval shrinks quickly. This will bound the area, but also make the height(and thus the sup) grow.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:21










  • $begingroup$
    @rubikscube09 Hmmm, ok. Would something like $f_n(x)=e^{nx}$ work?
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:24










  • $begingroup$
    @rubikscube09 I think the integral on that would grow instead of decay though hmmm... I'll keep trying! Thanks for the suggestion!
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:26








  • 1




    $begingroup$
    Do you know about indicator functions? They might be useful for something like this.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:28






  • 2




    $begingroup$
    Try to see how you could represent thin spikes that keep getting thinner and taller with indicator functions.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:34














1












1








1





$begingroup$


Problem Statement.



Good evening. As the title suggests, I am having a hard time with the following exercise:




Prove that there exists a sequence $f_1,f_2,ldots$ of functions on
$mathcal{L}^1([0,1])$ such that $lim_{nrightarrow
infty}||f_n||_1=0$ but $$sup{f_n(x): ninmathbf{Z}^+}=infty$$
for every $xin [0,1].$





Notation.



Here $mathcal{L}^1([0,1])$ means $mathcal{L}^1(lambda_{[0,1]})$ where $lambda_{[0,1]}$ represents Lebesgue measure restricted to the Borel subsets of $mathbf{R}$ that are contained in $[0,1]$.




Questions.



Ok, so my first question is how do I interpret $lim_{nrightarrow
infty}||f_n||_1=0$? I know that by definition $$||f||_1=int |f|,dmu$$ so I think this statement means $$lim_{nrightarrow infty}||f_n||=lim_{nrightarrowinfty}int|f_n|,dmu=0.$$ But I haven't been able to get anywhere with this, possibly because of my confusion with the other part of the question, which is even finding a sequence of functions that have the property $sup{f_n(x): ninmathbf{Z}^+}=infty$. I initially tried playing around with functions like $$f_n(x)=frac{1}{sqrt{nx}}$$ but this is undefined at $x=0$. So to summarize, my two questions are:




  1. Understanding the limit notation I pointed out.

  2. Finding a sequence of functions with an undefined supremum at every $x$.


Any suggestions towards either of these points of confusion would be wonderful.



Thank you for your time!






real-analysis sequences-and-series measure-theory lp-spaces supremum-and-infimum







share|cite|improve this question









$endgroup$




Problem Statement.



Good evening. As the title suggests, I am having a hard time with the following exercise:




Prove that there exists a sequence $f_1,f_2,ldots$ of functions on
$mathcal{L}^1([0,1])$ such that $lim_{nrightarrow
infty}||f_n||_1=0$ but $$sup{f_n(x): ninmathbf{Z}^+}=infty$$
for every $xin [0,1].$





Notation.



Here $mathcal{L}^1([0,1])$ means $mathcal{L}^1(lambda_{[0,1]})$ where $lambda_{[0,1]}$ represents Lebesgue measure restricted to the Borel subsets of $mathbf{R}$ that are contained in $[0,1]$.




Questions.



Ok, so my first question is how do I interpret $lim_{nrightarrow
infty}||f_n||_1=0$? I know that by definition $$||f||_1=int |f|,dmu$$ so I think this statement means $$lim_{nrightarrow infty}||f_n||=lim_{nrightarrowinfty}int|f_n|,dmu=0.$$ But I haven't been able to get anywhere with this, possibly because of my confusion with the other part of the question, which is even finding a sequence of functions that have the property $sup{f_n(x): ninmathbf{Z}^+}=infty$. I initially tried playing around with functions like $$f_n(x)=frac{1}{sqrt{nx}}$$ but this is undefined at $x=0$. So to summarize, my two questions are:




  1. Understanding the limit notation I pointed out.

  2. Finding a sequence of functions with an undefined supremum at every $x$.


Any suggestions towards either of these points of confusion would be wonderful.



Thank you for your time!






real-analysis sequences-and-series measure-theory lp-spaces supremum-and-infimum




real-analysis sequences-and-series measure-theory lp-spaces supremum-and-infimum






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 2 '18 at 5:16









Thy Art is MathThy Art is Math

489211




489211








  • 1




    $begingroup$
    Try a sequence of functions where the height of the functions grows, but the size of the interval shrinks quickly. This will bound the area, but also make the height(and thus the sup) grow.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:21










  • $begingroup$
    @rubikscube09 Hmmm, ok. Would something like $f_n(x)=e^{nx}$ work?
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:24










  • $begingroup$
    @rubikscube09 I think the integral on that would grow instead of decay though hmmm... I'll keep trying! Thanks for the suggestion!
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:26








  • 1




    $begingroup$
    Do you know about indicator functions? They might be useful for something like this.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:28






  • 2




    $begingroup$
    Try to see how you could represent thin spikes that keep getting thinner and taller with indicator functions.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:34














  • 1




    $begingroup$
    Try a sequence of functions where the height of the functions grows, but the size of the interval shrinks quickly. This will bound the area, but also make the height(and thus the sup) grow.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:21










  • $begingroup$
    @rubikscube09 Hmmm, ok. Would something like $f_n(x)=e^{nx}$ work?
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:24










  • $begingroup$
    @rubikscube09 I think the integral on that would grow instead of decay though hmmm... I'll keep trying! Thanks for the suggestion!
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:26








  • 1




    $begingroup$
    Do you know about indicator functions? They might be useful for something like this.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:28






  • 2




    $begingroup$
    Try to see how you could represent thin spikes that keep getting thinner and taller with indicator functions.
    $endgroup$
    – rubikscube09
    Dec 2 '18 at 5:34








1




1




$begingroup$
Try a sequence of functions where the height of the functions grows, but the size of the interval shrinks quickly. This will bound the area, but also make the height(and thus the sup) grow.
$endgroup$
– rubikscube09
Dec 2 '18 at 5:21




$begingroup$
Try a sequence of functions where the height of the functions grows, but the size of the interval shrinks quickly. This will bound the area, but also make the height(and thus the sup) grow.
$endgroup$
– rubikscube09
Dec 2 '18 at 5:21












$begingroup$
@rubikscube09 Hmmm, ok. Would something like $f_n(x)=e^{nx}$ work?
$endgroup$
– Thy Art is Math
Dec 2 '18 at 5:24




$begingroup$
@rubikscube09 Hmmm, ok. Would something like $f_n(x)=e^{nx}$ work?
$endgroup$
– Thy Art is Math
Dec 2 '18 at 5:24












$begingroup$
@rubikscube09 I think the integral on that would grow instead of decay though hmmm... I'll keep trying! Thanks for the suggestion!
$endgroup$
– Thy Art is Math
Dec 2 '18 at 5:26






$begingroup$
@rubikscube09 I think the integral on that would grow instead of decay though hmmm... I'll keep trying! Thanks for the suggestion!
$endgroup$
– Thy Art is Math
Dec 2 '18 at 5:26






1




1




$begingroup$
Do you know about indicator functions? They might be useful for something like this.
$endgroup$
– rubikscube09
Dec 2 '18 at 5:28




$begingroup$
Do you know about indicator functions? They might be useful for something like this.
$endgroup$
– rubikscube09
Dec 2 '18 at 5:28




2




2




$begingroup$
Try to see how you could represent thin spikes that keep getting thinner and taller with indicator functions.
$endgroup$
– rubikscube09
Dec 2 '18 at 5:34




$begingroup$
Try to see how you could represent thin spikes that keep getting thinner and taller with indicator functions.
$endgroup$
– rubikscube09
Dec 2 '18 at 5:34










1 Answer
1






active

oldest

votes


















2












$begingroup$

Here is an example of such a function:
$$
f_n:=ktimes 1_{[j/2^k,(j+1)/2^k]},
$$
where $n=2^k+j$ and $0le j< 2^k$.




$$
f_1equiv 0, f_2=1_{[0,1/2]}, f_3=1_{[1/2,1]} \
f_4=2times 1_{[0,1/4]}, f_5=2times 1_{[1/4,1/2]}, f_6=2times 1_{[1/2,3/4]}, f_7=2times 1_{[3/4,1]} \
ldots
$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Ok, the examples made that way clearer, thank you!
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:54











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%2f3022263%2fsequence-of-functions-on-mathcall10-1-with-lim-n-rightarrow-infty%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









2












$begingroup$

Here is an example of such a function:
$$
f_n:=ktimes 1_{[j/2^k,(j+1)/2^k]},
$$
where $n=2^k+j$ and $0le j< 2^k$.




$$
f_1equiv 0, f_2=1_{[0,1/2]}, f_3=1_{[1/2,1]} \
f_4=2times 1_{[0,1/4]}, f_5=2times 1_{[1/4,1/2]}, f_6=2times 1_{[1/2,3/4]}, f_7=2times 1_{[3/4,1]} \
ldots
$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Ok, the examples made that way clearer, thank you!
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:54
















2












$begingroup$

Here is an example of such a function:
$$
f_n:=ktimes 1_{[j/2^k,(j+1)/2^k]},
$$
where $n=2^k+j$ and $0le j< 2^k$.




$$
f_1equiv 0, f_2=1_{[0,1/2]}, f_3=1_{[1/2,1]} \
f_4=2times 1_{[0,1/4]}, f_5=2times 1_{[1/4,1/2]}, f_6=2times 1_{[1/2,3/4]}, f_7=2times 1_{[3/4,1]} \
ldots
$$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Ok, the examples made that way clearer, thank you!
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:54














2












2








2





$begingroup$

Here is an example of such a function:
$$
f_n:=ktimes 1_{[j/2^k,(j+1)/2^k]},
$$
where $n=2^k+j$ and $0le j< 2^k$.




$$
f_1equiv 0, f_2=1_{[0,1/2]}, f_3=1_{[1/2,1]} \
f_4=2times 1_{[0,1/4]}, f_5=2times 1_{[1/4,1/2]}, f_6=2times 1_{[1/2,3/4]}, f_7=2times 1_{[3/4,1]} \
ldots
$$






share|cite|improve this answer











$endgroup$



Here is an example of such a function:
$$
f_n:=ktimes 1_{[j/2^k,(j+1)/2^k]},
$$
where $n=2^k+j$ and $0le j< 2^k$.




$$
f_1equiv 0, f_2=1_{[0,1/2]}, f_3=1_{[1/2,1]} \
f_4=2times 1_{[0,1/4]}, f_5=2times 1_{[1/4,1/2]}, f_6=2times 1_{[1/2,3/4]}, f_7=2times 1_{[3/4,1]} \
ldots
$$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 2 '18 at 5:53

























answered Dec 2 '18 at 5:38









d.k.o.d.k.o.

8,667528




8,667528












  • $begingroup$
    Ok, the examples made that way clearer, thank you!
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:54


















  • $begingroup$
    Ok, the examples made that way clearer, thank you!
    $endgroup$
    – Thy Art is Math
    Dec 2 '18 at 5:54
















$begingroup$
Ok, the examples made that way clearer, thank you!
$endgroup$
– Thy Art is Math
Dec 2 '18 at 5:54




$begingroup$
Ok, the examples made that way clearer, thank you!
$endgroup$
– Thy Art is Math
Dec 2 '18 at 5:54


















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%2f3022263%2fsequence-of-functions-on-mathcall10-1-with-lim-n-rightarrow-infty%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

Quarter-circle Tiles

Image
up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2
Read more

build a pushdown automaton that recognizes the reverse language of a given pushdown automaton?

Image
0 I think it's similiar to NFAs. I replace the finite states of the given automaton for start-states for my new automaton. I do it with $epsilon$ -transition from the start state to the actual finite states of the given automaton. The start state of the given automaton will be my accepting(finite) state of my new one. All transitions will be reversed of course to recognize the reversed word. But how do I have to change the stack of the pushdown automaton? That's tricky. formal-languages automata context-free-grammar share | cite | improve this question edited Nov 29 '18 at 14:20 joee
Read more

Mont Emei

Image
Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog
Read more