Prove that convergence of the infinite series of integral of absolue values of a sequence of functions implies convergence
Let $(X,\Sigma,\mu)$ be a measure space and let $f_{n}:X\rightarrow \mathbb{R} $ is an integrable function such that $\sum_{n=1}^{\infty } \int |f_{n}|du$ is convergent.
Prove that $\sum_{n=1}^{\infty } f_{n}$ converges almost everywhere to an integrable function and that
$$\int \sum_{n=1}^{\infty}f_{n} du=\sum_{n=1}^{\infty}\int f_{n}du.$$
Noah Taylor
17
Answer
Answers can only be viewed under the following conditions:
- The questioner was satisfied with and accepted the answer, or
- The answer was evaluated as being 100% correct by the judge.
Daniel90
443
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 1622 views
- $40.00
Related Questions
- How do we take the mean of a mathematical function using statistics?
- Interior of union of two sets with empty interior
- Let $(X, ||\cdot||)$ be a normed space. Let $\{x_n\}$ and $\{y_n\}$ be two Cauchy sequences in X. Show that the seqience Show that the sequence $λ_n = ||x_n − y_n|| $ converges.
- real analysis
- Probability/Analysis Question
- real analysis
- Prove that $\frac{d \lambda}{d \mu} = \frac{d \lambda}{d \nu} \frac{d \nu}{d \mu}$ for $\sigma$-finite measures $\mu,\nu, \lambda$.
- Define $F : \mathbb{R}^ω → \mathbb{R}^ω$ by $F(x)_n = \sum^n_{k=1} x_k$. Determine whether $F$ restricts to give a well-defined map $F : (\ell_p, d_p) → (\ell_q, d_q)$