1
vote
0answers
34 views

Determining the sets of alpha for which some (Riemann, Lebesgue - integrals) exists

$$\int_0^{\infty} \frac{\sin(x)}{x^{\alpha}} \, dx.$$ $$\int_{[0, \infty]} \frac{\sin(x)}{x^{\alpha}} \, d \lambda(x).$$ $$\int_{\Bbb R^2} \frac{\sin(\| x \|)}{\| x \|^{\alpha}} \, d \lambda_2 ...
3
votes
1answer
45 views

Finding a dominating function for this sequence of functions

Problem: Find the limit $$\lim_{n\to\infty} \int_0^n \left( 1 + \frac{x}{n}\right )^{-n} \log(2 + \cos(x/n))dx$$ and justify your reasoning. My Solution: Let $f_n = \left( 1 + \frac{x}{n}\right ...
6
votes
3answers
113 views

Finding the integral of $\frac{x}{e^x + 1}$ [duplicate]

I've having some difficulty with finding this integral: $$ \int_0 ^{\infty} \frac{x}{e^x + 1}$$ Now usually I would use the monotone convergence theorem to write (using geometric series): $$f_n (x) ...
4
votes
3answers
176 views

General condition that Riemann and Lebesgue integrals are the same

I'd like to know that when Riemann integral and Lebesgue integral are the same in general. I know that a bounded Riemann integrable function on a closed interval is Lebesgue integrable and two ...
4
votes
1answer
113 views

Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0 $ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0 $ Can ...
0
votes
0answers
36 views

Can the theory of Lebesgue integration be extended in a way analogous to extending Riemann integrals to improper Riemann integrals?

I recently (last night) learned the definition of Lebesgue integration and one of the limitations I was told was that some improper Riemann integrals aren't Lebesgue integrable. It occurred to me ...
1
vote
2answers
73 views

if $f(x)$ is summable square function, then… [duplicate]

I have a question. If a normed function, that is to say $$ \int_{-\infty}^{\infty}|f(x)|^2dx<\infty~~~\text{(summable square function)}$$ then, $$\underset{\begin{array}{c} ...
2
votes
3answers
99 views

summable square function implies…?

I have difficulty to demonstrate this: $$ \int_{-\infty}^{\infty}|f(x)|^2dx<\infty~~~\text{(summable square function)}$$ then, $$\lim_{|x|\rightarrow\infty }f(x)=0$$ thank you.
3
votes
1answer
43 views

Conditions on a measure

Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>0$ $t^s\in L^1(\mathrm d\mu(t))$, but functions $\mathbf{1}_{t>0} $ and $\mathbf{1}_{t\in (0,1)}$ are not necessarily in ...
1
vote
2answers
110 views

Contradiction to Continuity of Integration?

For each of the two functions $f$ on $[1,\infty)$ defined below, show that $\lim_{n \rightarrow \infty} \int_1^n f$ exists while $f$ is not integrable over $[1,\infty)$. Does this contradict the ...
1
vote
2answers
127 views

prove $ F(x)=\int_0^\infty {\sin(tx)\over(t+1)\sqrt t} dt \in C^\infty(\mathbb R^*) $

prove that : $$ F(x)=\int_0^\infty {\sin(tx)\over(t+1)\sqrt t} \, dt \in C^\infty(\mathbb R^*) $$ i end up proving that $F(x)\in C^ \infty(\mathbb R^{*+})$ not $\mathbb R^*$ , and i studied the ...
0
votes
1answer
143 views

Prove convergence of improper integral using change of variable.

This may be trivial, but I could use some help... Consider a real function $f: (0,1) \rightarrow \mathbb{R}$, continuous, positive, but not necessarily bounded. Let $g: [0,1] \rightarrow [0,1]$ be a ...
5
votes
3answers
190 views

$ \int^{\infty}_0 |\frac{1}{(1+x)\sqrt x}|^p ~ \mathrm dx < \infty \implies p=?$

If $ f(x) = \frac{1}{(1+x)\sqrt x} $ how to find all $ p > 0 $ such that $$ \int^{\infty}_0 |f(x)|^p dx < \infty $$ The integral is with respect to lebesgue measure. Any solution or hints would ...
0
votes
1answer
102 views

The relation between arbitrary measure space and the Lebesgue integral

Let $(X, \mathcal F, \mu)$ be a measure space and $f\in M^+(X,\mu)$ (the measurable non-negative functions), and $t>0$. Now let $$S_f(t)=\{x\in X:f(x)>t\} \quad \Psi_f(t)=\mu(S_f(t))$$ Prove ...
1
vote
1answer
695 views

improper Riemann integral and Lebesgue integral

Let $f$ be a continuous function on $(0,1]$ and is defined as $f: [0,1] \to \mathbb R$. Show that if $f$ is lebesgue integrable on $[0,1]$, the improper Riemann integral $\lim_{\epsilon \to 0} ...