$int_{0}^{infty}frac{sin(x)}{x}dx$ exists, but $int_{mathbb{R^+}}frac{sin(x)}{x}dlambda$ doesn't exist?












1












$begingroup$


$int_{0}^{infty}frac{sin(x)}{x}dx$ is riemann integrable.



But how to prove that it's not Lebesgue integrable ?



(I tried contradiction).



If $f(x)=frac{sin(x)}{x}$, I supppose that $fin L^1(mathbb{R^+})$.



then, I' ll define the function sequence : $f_n(x)=frac{sin( x)}{x}mathbb{1}_{[0,n]}(x).$



We have $f_n$ converges $lambda$-a.e to $f$.



and $|f_n|leq|f|$, for all $1leq n$.



and $f$ integrable (hypothesis).



then with the dominated convergence theorem



$$lim_nint_{0}^{n}f_ndlambda=int_{mathbb{R^+}}fdlambda$$



which is equivalent to $$int_{0}^{infty}frac{sin(x)}{x}dx=int_{mathbb{R^+}}f
dlambda$$



Thats basically what I did.










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Hi Anas, as part of the community guidelines you need to provide some more context and background to your question. Furthermore if you have attempted the problem you need to provide your working. Just a heads up :-)
    $endgroup$
    – user150203
    Dec 23 '18 at 7:19










  • $begingroup$
    @DavidG, check .
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:31










  • $begingroup$
    Looks good to me! Just wanted to let you know. This page has very specific rules about posting. It annoyed me to start with, but once you get use to it you realise it's a great format to work with. Sorry I can't help you wing your question though.
    $endgroup$
    – user150203
    Dec 23 '18 at 7:35










  • $begingroup$
    Well, i can't find the contradiction with the last equality. but thanks anyway.
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:38










  • $begingroup$
    Actually it is not Riemann integrable either. It is only improperly Riemann integrable.
    $endgroup$
    – metamorphy
    Dec 23 '18 at 7:43
















1












$begingroup$


$int_{0}^{infty}frac{sin(x)}{x}dx$ is riemann integrable.



But how to prove that it's not Lebesgue integrable ?



(I tried contradiction).



If $f(x)=frac{sin(x)}{x}$, I supppose that $fin L^1(mathbb{R^+})$.



then, I' ll define the function sequence : $f_n(x)=frac{sin( x)}{x}mathbb{1}_{[0,n]}(x).$



We have $f_n$ converges $lambda$-a.e to $f$.



and $|f_n|leq|f|$, for all $1leq n$.



and $f$ integrable (hypothesis).



then with the dominated convergence theorem



$$lim_nint_{0}^{n}f_ndlambda=int_{mathbb{R^+}}fdlambda$$



which is equivalent to $$int_{0}^{infty}frac{sin(x)}{x}dx=int_{mathbb{R^+}}f
dlambda$$



Thats basically what I did.










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Hi Anas, as part of the community guidelines you need to provide some more context and background to your question. Furthermore if you have attempted the problem you need to provide your working. Just a heads up :-)
    $endgroup$
    – user150203
    Dec 23 '18 at 7:19










  • $begingroup$
    @DavidG, check .
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:31










  • $begingroup$
    Looks good to me! Just wanted to let you know. This page has very specific rules about posting. It annoyed me to start with, but once you get use to it you realise it's a great format to work with. Sorry I can't help you wing your question though.
    $endgroup$
    – user150203
    Dec 23 '18 at 7:35










  • $begingroup$
    Well, i can't find the contradiction with the last equality. but thanks anyway.
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:38










  • $begingroup$
    Actually it is not Riemann integrable either. It is only improperly Riemann integrable.
    $endgroup$
    – metamorphy
    Dec 23 '18 at 7:43














1












1








1





$begingroup$


$int_{0}^{infty}frac{sin(x)}{x}dx$ is riemann integrable.



But how to prove that it's not Lebesgue integrable ?



(I tried contradiction).



If $f(x)=frac{sin(x)}{x}$, I supppose that $fin L^1(mathbb{R^+})$.



then, I' ll define the function sequence : $f_n(x)=frac{sin( x)}{x}mathbb{1}_{[0,n]}(x).$



We have $f_n$ converges $lambda$-a.e to $f$.



and $|f_n|leq|f|$, for all $1leq n$.



and $f$ integrable (hypothesis).



then with the dominated convergence theorem



$$lim_nint_{0}^{n}f_ndlambda=int_{mathbb{R^+}}fdlambda$$



which is equivalent to $$int_{0}^{infty}frac{sin(x)}{x}dx=int_{mathbb{R^+}}f
dlambda$$



Thats basically what I did.










share|cite|improve this question











$endgroup$




$int_{0}^{infty}frac{sin(x)}{x}dx$ is riemann integrable.



But how to prove that it's not Lebesgue integrable ?



(I tried contradiction).



If $f(x)=frac{sin(x)}{x}$, I supppose that $fin L^1(mathbb{R^+})$.



then, I' ll define the function sequence : $f_n(x)=frac{sin( x)}{x}mathbb{1}_{[0,n]}(x).$



We have $f_n$ converges $lambda$-a.e to $f$.



and $|f_n|leq|f|$, for all $1leq n$.



and $f$ integrable (hypothesis).



then with the dominated convergence theorem



$$lim_nint_{0}^{n}f_ndlambda=int_{mathbb{R^+}}fdlambda$$



which is equivalent to $$int_{0}^{infty}frac{sin(x)}{x}dx=int_{mathbb{R^+}}f
dlambda$$



Thats basically what I did.







integration lebesgue-integral






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 23 '18 at 7:37







user150203

















asked Dec 23 '18 at 7:15









Anas BOUALIIAnas BOUALII

1408




1408








  • 1




    $begingroup$
    Hi Anas, as part of the community guidelines you need to provide some more context and background to your question. Furthermore if you have attempted the problem you need to provide your working. Just a heads up :-)
    $endgroup$
    – user150203
    Dec 23 '18 at 7:19










  • $begingroup$
    @DavidG, check .
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:31










  • $begingroup$
    Looks good to me! Just wanted to let you know. This page has very specific rules about posting. It annoyed me to start with, but once you get use to it you realise it's a great format to work with. Sorry I can't help you wing your question though.
    $endgroup$
    – user150203
    Dec 23 '18 at 7:35










  • $begingroup$
    Well, i can't find the contradiction with the last equality. but thanks anyway.
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:38










  • $begingroup$
    Actually it is not Riemann integrable either. It is only improperly Riemann integrable.
    $endgroup$
    – metamorphy
    Dec 23 '18 at 7:43














  • 1




    $begingroup$
    Hi Anas, as part of the community guidelines you need to provide some more context and background to your question. Furthermore if you have attempted the problem you need to provide your working. Just a heads up :-)
    $endgroup$
    – user150203
    Dec 23 '18 at 7:19










  • $begingroup$
    @DavidG, check .
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:31










  • $begingroup$
    Looks good to me! Just wanted to let you know. This page has very specific rules about posting. It annoyed me to start with, but once you get use to it you realise it's a great format to work with. Sorry I can't help you wing your question though.
    $endgroup$
    – user150203
    Dec 23 '18 at 7:35










  • $begingroup$
    Well, i can't find the contradiction with the last equality. but thanks anyway.
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:38










  • $begingroup$
    Actually it is not Riemann integrable either. It is only improperly Riemann integrable.
    $endgroup$
    – metamorphy
    Dec 23 '18 at 7:43








1




1




$begingroup$
Hi Anas, as part of the community guidelines you need to provide some more context and background to your question. Furthermore if you have attempted the problem you need to provide your working. Just a heads up :-)
$endgroup$
– user150203
Dec 23 '18 at 7:19




$begingroup$
Hi Anas, as part of the community guidelines you need to provide some more context and background to your question. Furthermore if you have attempted the problem you need to provide your working. Just a heads up :-)
$endgroup$
– user150203
Dec 23 '18 at 7:19












$begingroup$
@DavidG, check .
$endgroup$
– Anas BOUALII
Dec 23 '18 at 7:31




$begingroup$
@DavidG, check .
$endgroup$
– Anas BOUALII
Dec 23 '18 at 7:31












$begingroup$
Looks good to me! Just wanted to let you know. This page has very specific rules about posting. It annoyed me to start with, but once you get use to it you realise it's a great format to work with. Sorry I can't help you wing your question though.
$endgroup$
– user150203
Dec 23 '18 at 7:35




$begingroup$
Looks good to me! Just wanted to let you know. This page has very specific rules about posting. It annoyed me to start with, but once you get use to it you realise it's a great format to work with. Sorry I can't help you wing your question though.
$endgroup$
– user150203
Dec 23 '18 at 7:35












$begingroup$
Well, i can't find the contradiction with the last equality. but thanks anyway.
$endgroup$
– Anas BOUALII
Dec 23 '18 at 7:38




$begingroup$
Well, i can't find the contradiction with the last equality. but thanks anyway.
$endgroup$
– Anas BOUALII
Dec 23 '18 at 7:38












$begingroup$
Actually it is not Riemann integrable either. It is only improperly Riemann integrable.
$endgroup$
– metamorphy
Dec 23 '18 at 7:43




$begingroup$
Actually it is not Riemann integrable either. It is only improperly Riemann integrable.
$endgroup$
– metamorphy
Dec 23 '18 at 7:43










1 Answer
1






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$begingroup$

Hint. If such function is Lebesgue integrable then also its absolute value is Lebesgue integrable. Let $n$ a positive integer, then
$$
int_0^{2pi n} frac{|sin x|}{x} ,dx = sum_{k=0}^{n-1} int_{2pi k}^{2pi(k+1)} frac{|sin x|}{x},dx
ge sum_{k=0}^{n-1} frac{1}{2pi (k+1)} int_{0}^{2pi} |sin x|,dx
$$

Can you take it from here?






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Yes, I get it. thank you. I knew the proposition , but i didn't know how to use it correctly.
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:47












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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









1












$begingroup$

Hint. If such function is Lebesgue integrable then also its absolute value is Lebesgue integrable. Let $n$ a positive integer, then
$$
int_0^{2pi n} frac{|sin x|}{x} ,dx = sum_{k=0}^{n-1} int_{2pi k}^{2pi(k+1)} frac{|sin x|}{x},dx
ge sum_{k=0}^{n-1} frac{1}{2pi (k+1)} int_{0}^{2pi} |sin x|,dx
$$

Can you take it from here?






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Yes, I get it. thank you. I knew the proposition , but i didn't know how to use it correctly.
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:47
















1












$begingroup$

Hint. If such function is Lebesgue integrable then also its absolute value is Lebesgue integrable. Let $n$ a positive integer, then
$$
int_0^{2pi n} frac{|sin x|}{x} ,dx = sum_{k=0}^{n-1} int_{2pi k}^{2pi(k+1)} frac{|sin x|}{x},dx
ge sum_{k=0}^{n-1} frac{1}{2pi (k+1)} int_{0}^{2pi} |sin x|,dx
$$

Can you take it from here?






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Yes, I get it. thank you. I knew the proposition , but i didn't know how to use it correctly.
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:47














1












1








1





$begingroup$

Hint. If such function is Lebesgue integrable then also its absolute value is Lebesgue integrable. Let $n$ a positive integer, then
$$
int_0^{2pi n} frac{|sin x|}{x} ,dx = sum_{k=0}^{n-1} int_{2pi k}^{2pi(k+1)} frac{|sin x|}{x},dx
ge sum_{k=0}^{n-1} frac{1}{2pi (k+1)} int_{0}^{2pi} |sin x|,dx
$$

Can you take it from here?






share|cite|improve this answer









$endgroup$



Hint. If such function is Lebesgue integrable then also its absolute value is Lebesgue integrable. Let $n$ a positive integer, then
$$
int_0^{2pi n} frac{|sin x|}{x} ,dx = sum_{k=0}^{n-1} int_{2pi k}^{2pi(k+1)} frac{|sin x|}{x},dx
ge sum_{k=0}^{n-1} frac{1}{2pi (k+1)} int_{0}^{2pi} |sin x|,dx
$$

Can you take it from here?







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 23 '18 at 7:40









Robert ZRobert Z

102k1072145




102k1072145












  • $begingroup$
    Yes, I get it. thank you. I knew the proposition , but i didn't know how to use it correctly.
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:47


















  • $begingroup$
    Yes, I get it. thank you. I knew the proposition , but i didn't know how to use it correctly.
    $endgroup$
    – Anas BOUALII
    Dec 23 '18 at 7:47
















$begingroup$
Yes, I get it. thank you. I knew the proposition , but i didn't know how to use it correctly.
$endgroup$
– Anas BOUALII
Dec 23 '18 at 7:47




$begingroup$
Yes, I get it. thank you. I knew the proposition , but i didn't know how to use it correctly.
$endgroup$
– Anas BOUALII
Dec 23 '18 at 7:47


















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