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Let $f:\Bbb R\to \Bbb R$ be defined by

$f(x)$=\begin{cases} \frac{1}{\sqrt x};\text{$0

Define $g:\Bbb R\to \Bbb R$ by $g(x)=\sum_{n=1}^\infty \dfrac{1}{2^n}f(x-r_n)$ where $\{r_n\}_n$ is an enumeration of rationals in $\Bbb R$.

Show that $f,g$ is integrable.

Since $f$ is unbounded we need to talk about Lebesgue-Integrability here.

Now $\int f=2\sqrt x|_0^1=2<\infty \implies f$ is Lebesgue Integrable.

I am unable to show this for $g$. Any help.

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    Why do we need to talk about Lebesgue integrability? $f$ is Riemann Integrable imo. As you did, you took the limit for $r \to 0$ of the Riemann Integrals in $(r,1)$. – 2017-01-27
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    But $f$ is not bounded ? As $x\to 0\implies f(x)\to \infty$ – 2017-01-27
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    I don't think this is a problem becuase you integrate in $[R,1]$ and then take the limit. – 2017-01-27
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    "Lebesgue's integrability has nothing to do with the Lebesgue integral" cit wikipedia, my bad! :) You can call it this way if you please. – 2017-01-27

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Use Tonelli: For non negative measurable $g_n$, we have $\int \sum_n g_n = \sum_n \int g_n $, so if we let $g_n(x) = {1 \over 2^n} f(x-r_n)$, we see that $\int g = \int \sum_n g_n = \sum_n \int g_n = \sum_n 2 {1 \over 2^n} < \infty$.

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    What is Tonelli? – 2017-01-27
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    A powerful alcoholic drink. See https://en.wikipedia.org/wiki/Fubini's_theorem. I will see if I can get a more apropos link. – 2017-01-27
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    @copper.hat Is this not a consequence of monotone convergence? – 2017-01-27
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    @Math1000: Yes, that would be one way of doing it. I was thinking of interchanging the order of integration (using the counting measure for one), but the monotone convergence theorem would be more straightforward. – 2017-01-27
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    @copper.hat; I don't know about Tonelli nor could I understand it from Wiki;Is it possible to do it in some other easier way – 2017-01-27
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    I read the first 3 chap. of Royden ;but there nothing is explained about Tonelli;are you reading some other book – 2017-01-27
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    See the comment from Math1000, the monotone convergence theorem is a good approach. I don't have access to a copy of Royden's book at the moment, but I would be surprised if there is no version of Tonelli's theorem there; it might be under Fubini or product measures. In any case, the purpose is to justify the interchange of integral & summation. – 2017-01-27