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

Question

‎Let‎ ‎$‎‎f:‎\mathbb{R}\to‎\mathbb{R}‎$ 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 ‎$‎f:‎\mathbb{R}\to‎\mathbb{R}‎$ ‎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 ‎$‎‎\lim_{x\to +‎\infty‎}f(x)=0$ true?

See https://math.stackexchange.com/questions/829927/general-condition-that-riemann-and-lebesgue-integrals-are-the-same — M.H.Hooshmand, Dec 20 '18 at 09:32

Kavi Rama Murthy · Answer 1 · 2018-12-20T09:09:30.033

2

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 $f(x)=c\sum nI_{(n,n+\frac 1 {n^{3}})}$ where $c$ is chosen such that $f$ integrates to $1$.

edited Dec 20 '18 at 09:09

answered Dec 20 '18 at 08:46

Kavi Rama Murthy

311,013

1 and 2 you mean that Question‎(1) and Question‎(2) or the both are related Q2? – soodehMehboodi Dec 20 '18 at 09:07
@soodehMehboodi I hope the answer is clear now. – Kavi Rama Murthy Dec 20 '18 at 09:10
@ Kavi Rama Murthy, thanks a lot. – soodehMehboodi Dec 20 '18 at 09:38
@ Kavi Rama Murthy, would you tell me, which conditions make question 2) be true? – soodehMehboodi Dec 20 '18 at 09:51
1

@soodehMehboodi If a density function is uniformly continuous on $\mathbb R$ we can show that $f(x) \to 0$ as $x \to \infty$. – Kavi Rama Murthy Dec 20 '18 at 09:55
@ Kavi Rama Murthy, Excuse me , I have a general question, If $F$ is the CDF function of the PDF function $f$ then Is $F^\prime=f$ always true? would you help me? thanks. – soodehMehboodi Dec 20 '18 at 15:44
@soodehMehboodi $F'=f$ almost everywhere for any PDF. – Kavi Rama Murthy Dec 20 '18 at 23:10
@ Kavi Rama Murthy, Thank you. – soodehMehboodi Dec 21 '18 at 06:41
@ Kavi Rama Murthy, Would you tell me that, why it must be uniformly continuous? Can continuous work alone? – soodehMehboodi Dec 26 '18 at 06:20
Draw triangles with vertices $(n-\frac 1 {n^{2}},0), (n+\frac 1 {n{2}},0), (n,1)$ (n>2) and combine these to get the graph of a continuous function $f$. A constant multiple of this function is a density function which is continuous but its vales at $3,4,\cdots$ do not tend to $0$. – Kavi Rama Murthy Dec 26 '18 at 06:27
@ Kavi Rama Murthy, Firstly thanks a lot for your leading, Now my question is: " for a probability density function ($f$), does there exist a cumulative distribution function ($F$)"? – soodehMehboodi Dec 30 '18 at 10:38
@soodehMehboodi Of course! $F(x)=\int_{-\infty}^{x} f(t), dt$ defines the cumulative distribution function. – Kavi Rama Murthy Dec 30 '18 at 11:33
@ Kavi Rama Murthy, Hi again, Excuse me for my several questions, I have a question about your previous comment. if $f$ has a continuous derivative then, is question 2 true? I mean that does continuous derivative work instead of uniformly continuous? – soodehMehboodi Dec 31 '18 at 08:01
@soodehMehboodi No. You an construct infinitely differentiable density functions that do not tend to $0$. – Kavi Rama Murthy Dec 31 '18 at 08:10
@ Kavi Rama Murthy, Thanks for your advice, also quickly responding. – soodehMehboodi Dec 31 '18 at 08:14
@ Kavi Rama Murthy, hello, I know that if $f$ is continuous then $F^\prime=f$ is always true. Now I have a question : if $f$ is monotonic, does the equality always true or a.e? – soodehMehboodi Jan 10 '19 at 09:29
@soodehMehboodi Yes. Lebesgue's Theorem says that if $f$ is integrable on finite intervals and $F(x)=F(a)+\int_{a}^{x}f(t), dt$ then $F'(x)=f(x)$ almost everywhere. – Kavi Rama Murthy Jan 10 '19 at 09:32
@ Kavi Rama Murthy, you mean that monotonic can not work same as continuous, and the equality is true a,e. not always. – soodehMehboodi Jan 10 '19 at 09:36

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

1 Answers1