Tagged Questions
0
votes
0answers
64 views
Show $f$ is in $L^1$ (d$\mu$) space and $_X\int f $ d$\mu=\lim_{n\to \infty}\int_X f_n d\mu$
Suppose $f$ is in $L^1$($\mu$). Prove that for each $\epsilon > 0$ , there exists a $\delta > 0$ so that the $\int |f|\mathrm d\mu$ < $\epsilon$ over the set $E$ whenever $\mu(E) < ...
2
votes
2answers
85 views
Solve limits in Lebesgue integral
Solve the limits of below:
(1) $\lim\limits_{n \to \infty} \int_0^n (1+\frac{x}{n})^n e^{-2x}dx$.
(2) $\lim\limits_{n \to \infty} \int_0^n (1-\frac{x}{n})^n e^{\frac{x}{2}}dx$.
(3) ...
2
votes
1answer
47 views
The Lebesgue Theory basic Application , get stuck
Ok, I am working on a very easy question but I get stuck when I trying to justify my answer.
I know that, in order to use Lebesgue's dominated Convergence Theorem, there are two conditions that we ...
1
vote
0answers
70 views
$\lim_{n \to \infty} \int^n_{-n}fdm=\int fdm$
Let $f:\mathbb{R} \to \mathbb{R}$ such that $f$ is integrable over $[-n,n]$ for every $n \in \mathbb{R}$ and assume that
$$\lim_{n \to \infty} \int^n_{-n}fdm < \infty.$$
Proposition: $f$ is ...
