Notes - Analysis III TT23, Integration and limits
Flashcards
Can you state the theorem that relates uniform convergence and integration?
Suppose that $f _ n : [a, b] \to \mathbb R$ is integrable and that $f _ n \to f$ uniformly. Then $f$ is also integrable, and
\[\lim_{n \to \infty} \int^b_a f_n = \int^b_a f = \int^b_a \lim_{n \to \infty} f_n\]Can you state the theorem that lets you do the following
\[\int^b_a \sum_{i} \phi_i = \sum_i \int^b_a \phi_i\]
?
Suppose that $\phi _ i : [a, b] \to \mathbb R$ for $i=1,\ldots,$ are integrable and that $ \vert \phi _ i(x) \vert \le M _ i$ for all $x\in[a, b]$ and $\sum _ {i=1}^\infty M _ i$ converges. Then the sum is integrable and
\[\int^b_a \sum_{i} \phi_i = \sum_i \int^b_a \phi_i\]Proofs
Prove that if $f _ n : [a, b] \to \mathbb R$ is integrable and that $f _ n \to f$ uniformly, then $f$ is also integrable, and
\[\lim_{n \to \infty} \int^b_a f_n = \int^b_a f = \int^b_a \lim_{n \to \infty} f_n\]
Todo (analysis iii, page 29).