MAT6 -- Spring 2013

Integration and Fourier Theory

7. lecture

Monday, February 25,  08:15
Place: grupperum.

The program for today

08:15-12:00
Start by reading Fatou's Lemma and the Lebesgue Dominated Convergence Theorem.

Exercise 7, page 32. Hint: define $g_n=f_1-f_n$ and use Lebesgue's monotone convergence theorem.

Exercise 8, page 32. Hint: let $x$ be an arbitrary point. If $x\in E$, then $f_n(x)=[(-1)^{n+1}+1]/2$; compute $\liminf f_n(x)$. If $x\not\in E$ then $f_n(x)=[(-1)^{n}+1]/2$; compute again $\liminf f_n(x)$. Then compute $\liminf \int_{X} f_n d\mu$ and check Fatou's Lemma.

Exercise 9, page 32. Extra hint: remember that $\lim_{x\to 0}\frac{\ln(1+x)}{x}=1$.

Written by Horia Cornean.