‎Is ‎$‎int_{-‎infty‎}^{+infty}f(x)dx‎$ ‎convergent (as Riemann integrable), where ‎$‎f$ is...












1












$begingroup$


‎Let‎ ‎$‎‎f:‎mathbb{R}to‎mathbb{R}‎$ be a Riemann integrable probability density function on every closed ‎interval ‎of ‎‎$‎‎‎mathbb{R}‎$‎. As I explained in my previous question (Does there exist any probability density function ‎$‎f:‎mathbb{R}to‎mathbb{R}‎$ ‎which is not Riemann integrable?) , Now I have 2 questions as follows:



Question‎(1)‎‎‎‎ ‎Is ‎‎$‎‎int_{-‎infty‎}^{+infty}f(x)dx=‎‎int_{-‎infty‎}^{0}f(x)dx+‎‎int_{0‎}^{+infty}f(x)dx‎‎$ ‎convergent?‎



Question‎‎‎(2) Is ‎$‎‎lim_{xto +‎infty‎}f(x)=0$ true?






probability integration probability-theory probability-distributions







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  • 1




    $begingroup$
    See math.stackexchange.com/questions/829927/…
    $endgroup$
    – M.H.Hooshmand
    Dec 20 '18 at 9:32
















1












$begingroup$


‎Let‎ ‎$‎‎f:‎mathbb{R}to‎mathbb{R}‎$ be a Riemann integrable probability density function on every closed ‎interval ‎of ‎‎$‎‎‎mathbb{R}‎$‎. As I explained in my previous question (Does there exist any probability density function ‎$‎f:‎mathbb{R}to‎mathbb{R}‎$ ‎which is not Riemann integrable?) , Now I have 2 questions as follows:



Question‎(1)‎‎‎‎ ‎Is ‎‎$‎‎int_{-‎infty‎}^{+infty}f(x)dx=‎‎int_{-‎infty‎}^{0}f(x)dx+‎‎int_{0‎}^{+infty}f(x)dx‎‎$ ‎convergent?‎



Question‎‎‎(2) Is ‎$‎‎lim_{xto +‎infty‎}f(x)=0$ true?






probability integration probability-theory probability-distributions







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$endgroup$








  • 1




    $begingroup$
    See math.stackexchange.com/questions/829927/…
    $endgroup$
    – M.H.Hooshmand
    Dec 20 '18 at 9:32














1












1








1





$begingroup$


‎Let‎ ‎$‎‎f:‎mathbb{R}to‎mathbb{R}‎$ be a Riemann integrable probability density function on every closed ‎interval ‎of ‎‎$‎‎‎mathbb{R}‎$‎. As I explained in my previous question (Does there exist any probability density function ‎$‎f:‎mathbb{R}to‎mathbb{R}‎$ ‎which is not Riemann integrable?) , Now I have 2 questions as follows:



Question‎(1)‎‎‎‎ ‎Is ‎‎$‎‎int_{-‎infty‎}^{+infty}f(x)dx=‎‎int_{-‎infty‎}^{0}f(x)dx+‎‎int_{0‎}^{+infty}f(x)dx‎‎$ ‎convergent?‎



Question‎‎‎(2) Is ‎$‎‎lim_{xto +‎infty‎}f(x)=0$ true?






probability integration probability-theory probability-distributions







share|cite|improve this question









$endgroup$




‎Let‎ ‎$‎‎f:‎mathbb{R}to‎mathbb{R}‎$ be a Riemann integrable probability density function on every closed ‎interval ‎of ‎‎$‎‎‎mathbb{R}‎$‎. As I explained in my previous question (Does there exist any probability density function ‎$‎f:‎mathbb{R}to‎mathbb{R}‎$ ‎which is not Riemann integrable?) , Now I have 2 questions as follows:



Question‎(1)‎‎‎‎ ‎Is ‎‎$‎‎int_{-‎infty‎}^{+infty}f(x)dx=‎‎int_{-‎infty‎}^{0}f(x)dx+‎‎int_{0‎}^{+infty}f(x)dx‎‎$ ‎convergent?‎



Question‎‎‎(2) Is ‎$‎‎lim_{xto +‎infty‎}f(x)=0$ true?






probability integration probability-theory probability-distributions




probability integration probability-theory probability-distributions






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asked Dec 20 '18 at 8:39









soodehMehboodisoodehMehboodi

62138




62138








  • 1




    $begingroup$
    See math.stackexchange.com/questions/829927/…
    $endgroup$
    – M.H.Hooshmand
    Dec 20 '18 at 9:32














  • 1




    $begingroup$
    See math.stackexchange.com/questions/829927/…
    $endgroup$
    – M.H.Hooshmand
    Dec 20 '18 at 9:32








1




1




$begingroup$
See math.stackexchange.com/questions/829927/…
$endgroup$
– M.H.Hooshmand
Dec 20 '18 at 9:32




$begingroup$
See math.stackexchange.com/questions/829927/…
$endgroup$
– M.H.Hooshmand
Dec 20 '18 at 9:32










1 Answer
1






active

oldest

votes


















2












$begingroup$

For question 1) the answer is YES and it follows from the fact that the Riemann integral of a Riemann integrable fucntion coincides with the Lebesgue integral.
For question 2) the answer is NO: let $f(x)=csum nI_{(n,n+frac 1 {n^{3}})}$ where $c$ is chosen such that $f$ integrates to $1$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    1 and 2 you mean that Question‎(1) and Question‎(2) or the both are related Q2?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:07










  • $begingroup$
    @soodehMehboodi I hope the answer is clear now.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:10










  • $begingroup$
    @ Kavi Rama Murthy, thanks a lot.
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:38










  • $begingroup$
    @ Kavi Rama Murthy, would you tell me, which conditions make question 2) be true?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:51






  • 1




    $begingroup$
    @soodehMehboodi If a density function is uniformly continuous on $mathbb R$ we can show that $f(x) to 0$ as $x to infty$.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:55











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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

For question 1) the answer is YES and it follows from the fact that the Riemann integral of a Riemann integrable fucntion coincides with the Lebesgue integral.
For question 2) the answer is NO: let $f(x)=csum nI_{(n,n+frac 1 {n^{3}})}$ where $c$ is chosen such that $f$ integrates to $1$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    1 and 2 you mean that Question‎(1) and Question‎(2) or the both are related Q2?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:07










  • $begingroup$
    @soodehMehboodi I hope the answer is clear now.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:10










  • $begingroup$
    @ Kavi Rama Murthy, thanks a lot.
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:38










  • $begingroup$
    @ Kavi Rama Murthy, would you tell me, which conditions make question 2) be true?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:51






  • 1




    $begingroup$
    @soodehMehboodi If a density function is uniformly continuous on $mathbb R$ we can show that $f(x) to 0$ as $x to infty$.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:55
















2












$begingroup$

For question 1) the answer is YES and it follows from the fact that the Riemann integral of a Riemann integrable fucntion coincides with the Lebesgue integral.
For question 2) the answer is NO: let $f(x)=csum nI_{(n,n+frac 1 {n^{3}})}$ where $c$ is chosen such that $f$ integrates to $1$.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    1 and 2 you mean that Question‎(1) and Question‎(2) or the both are related Q2?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:07










  • $begingroup$
    @soodehMehboodi I hope the answer is clear now.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:10










  • $begingroup$
    @ Kavi Rama Murthy, thanks a lot.
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:38










  • $begingroup$
    @ Kavi Rama Murthy, would you tell me, which conditions make question 2) be true?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:51






  • 1




    $begingroup$
    @soodehMehboodi If a density function is uniformly continuous on $mathbb R$ we can show that $f(x) to 0$ as $x to infty$.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:55














2












2








2





$begingroup$

For question 1) the answer is YES and it follows from the fact that the Riemann integral of a Riemann integrable fucntion coincides with the Lebesgue integral.
For question 2) the answer is NO: let $f(x)=csum nI_{(n,n+frac 1 {n^{3}})}$ where $c$ is chosen such that $f$ integrates to $1$.






share|cite|improve this answer











$endgroup$



For question 1) the answer is YES and it follows from the fact that the Riemann integral of a Riemann integrable fucntion coincides with the Lebesgue integral.
For question 2) the answer is NO: let $f(x)=csum nI_{(n,n+frac 1 {n^{3}})}$ where $c$ is chosen such that $f$ integrates to $1$.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 20 '18 at 9:09

























answered Dec 20 '18 at 8:46









Kavi Rama MurthyKavi Rama Murthy

62.9k42362




62.9k42362












  • $begingroup$
    1 and 2 you mean that Question‎(1) and Question‎(2) or the both are related Q2?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:07










  • $begingroup$
    @soodehMehboodi I hope the answer is clear now.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:10










  • $begingroup$
    @ Kavi Rama Murthy, thanks a lot.
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:38










  • $begingroup$
    @ Kavi Rama Murthy, would you tell me, which conditions make question 2) be true?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:51






  • 1




    $begingroup$
    @soodehMehboodi If a density function is uniformly continuous on $mathbb R$ we can show that $f(x) to 0$ as $x to infty$.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:55


















  • $begingroup$
    1 and 2 you mean that Question‎(1) and Question‎(2) or the both are related Q2?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:07










  • $begingroup$
    @soodehMehboodi I hope the answer is clear now.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:10










  • $begingroup$
    @ Kavi Rama Murthy, thanks a lot.
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:38










  • $begingroup$
    @ Kavi Rama Murthy, would you tell me, which conditions make question 2) be true?
    $endgroup$
    – soodehMehboodi
    Dec 20 '18 at 9:51






  • 1




    $begingroup$
    @soodehMehboodi If a density function is uniformly continuous on $mathbb R$ we can show that $f(x) to 0$ as $x to infty$.
    $endgroup$
    – Kavi Rama Murthy
    Dec 20 '18 at 9:55
















$begingroup$
1 and 2 you mean that Question‎(1) and Question‎(2) or the both are related Q2?
$endgroup$
– soodehMehboodi
Dec 20 '18 at 9:07




$begingroup$
1 and 2 you mean that Question‎(1) and Question‎(2) or the both are related Q2?
$endgroup$
– soodehMehboodi
Dec 20 '18 at 9:07












$begingroup$
@soodehMehboodi I hope the answer is clear now.
$endgroup$
– Kavi Rama Murthy
Dec 20 '18 at 9:10




$begingroup$
@soodehMehboodi I hope the answer is clear now.
$endgroup$
– Kavi Rama Murthy
Dec 20 '18 at 9:10












$begingroup$
@ Kavi Rama Murthy, thanks a lot.
$endgroup$
– soodehMehboodi
Dec 20 '18 at 9:38




$begingroup$
@ Kavi Rama Murthy, thanks a lot.
$endgroup$
– soodehMehboodi
Dec 20 '18 at 9:38












$begingroup$
@ Kavi Rama Murthy, would you tell me, which conditions make question 2) be true?
$endgroup$
– soodehMehboodi
Dec 20 '18 at 9:51




$begingroup$
@ Kavi Rama Murthy, would you tell me, which conditions make question 2) be true?
$endgroup$
– soodehMehboodi
Dec 20 '18 at 9:51




1




1




$begingroup$
@soodehMehboodi If a density function is uniformly continuous on $mathbb R$ we can show that $f(x) to 0$ as $x to infty$.
$endgroup$
– Kavi Rama Murthy
Dec 20 '18 at 9:55




$begingroup$
@soodehMehboodi If a density function is uniformly continuous on $mathbb R$ we can show that $f(x) to 0$ as $x to infty$.
$endgroup$
– Kavi Rama Murthy
Dec 20 '18 at 9:55


















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