Tagged Questions

0
votes
1answer
50 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
112 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
56 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
267 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} ...