All Questions
32
questions
3
votes
3
answers
125
views
Convergence a.e. of $\sum_{n=1}^{\infty} \frac{a_n}{\sqrt{\left|x-r_n\right|}}$, where $\sum_{n=1}^{\infty}\left|a_n\right|<\infty$
Let $\left\{a_n\right\}_{n=1}^{\infty}$ and $\left\{r_n\right\}_{n=1}^{\infty}$ two sequences of real numbers, and suppose that: (i) $\sum_{n=1}^{\infty}\left|a_n\right|<\infty$; (ii) $r_i \neq r_j$...
0
votes
0
answers
74
views
Alternate Proof of the Monotone Convergence Theorem
I am following Folland’s Measure Theory book, and it contains a proof for the MCT that uses the fact that the integral of a simple function induces a measure, and here is my proof without that fact.
...
- 436
0
votes
2
answers
376
views
If $\sum^\infty_{n=1} \int |f_n| d\mu<\infty$, then $\sum_{n=1}^\infty f_n(x) d\mu$ converges to $f(x)$ , where $f_n$ and $f$ are integrable?
Want to prove that if $\{f_n\}$ is a sequence of integrable functions and $\sum^\infty_{n=1} \int |f_n| d\mu<\infty$, then $\sum_{n=1}^\infty f_n(x) d\mu$ is convergent to integrable function $f(x)$...
- 520
0
votes
1
answer
69
views
If $f$ is integrable, how to show that $\sum_{n=1}^\infty 2^n\lambda(\{\lvert f\rvert\ge 2^n\})<\infty$ [duplicate]
I want to show the following converse of an (apparently) common integration theory exercise.
Suppose $f:[0,1]\rightarrow\mathbb R$ is integrable, then
$$\sum_{n=1}^\infty 2^n\lambda(\{\lvert f\rvert\...
- 2,276
0
votes
0
answers
30
views
Interchange Lebesgue-Integral and infinite Sum with $f_n=\chi_{[n,n+1)}-\chi_{[n+1,n+2)}$
Let $f_n \colon \mathbb{R} \to \mathbb{R}$, $f_n = \chi_{[n,n+1)}-\chi_{[n+1,n+2)}$ where $\chi$ is the indicator function. Have I calculated the following statements correctly? $\lambda$ is the ...
- 387
1
vote
1
answer
71
views
Folland’s Real Analysis, Q2.13
I have been working through this proof and came across this link. The original problem can be found in that link as well as Exercise 2. My question is in the middle of the second page they claim the ...
- 6,277
3
votes
1
answer
339
views
Limit of integrals where both bounds and integrand depend on $n$
I want to find the limit
$$\lim_{n\rightarrow\infty}\int_{[0,n]}\left(1+\frac{x}{n}\right)^n e^{-2x} \, d\lambda(x)$$
I have noted the following: It is well known that $(1+\frac{x}{n})^n$ converges ...
- 2,276
1
vote
1
answer
202
views
Let $\mu_n$ be measures and $\mu=\sum_{n=1}^\infty \mu_n$. Show for measurable, positive $f$: $\int_Xf\ d\mu = \sum\int_X f\ d\mu_n$
Let $(X,\mathscr{S})$ be a measurable space, $\mu_n$ be measures and $\mu=\sum_{n=1}^\infty \mu_n$. I want to show for measurable $f:X\rightarrow[0,\infty]$:
$$\int_Xf\ d\mu = \sum_{n=1}^\infty\int_X ...
- 2,276
1
vote
1
answer
43
views
Using the dominated convergence theorem to show the following equality
I am trying to show the above using the Dominated convergence theorem. I have let the integrand be the limit point of a sequence of functions obtained by replacing the log by its expansion ($\log(\...
- 1,285
2
votes
2
answers
143
views
Proving an integral is equal to $\sum_{k=1}^\infty\frac{1}{(p+k)^2}$ for $p>0$.
I'm studying for a qualifying exam and I'm stuck on this problem from Bass's "Real Analysis for Graduate Students" (Exercise 7.14). It asks us to prove that
$$\sum_{k=1}^\infty\frac{1}{(p+k)^2}=-\...
- 1,054
1
vote
1
answer
76
views
Clarification in swapping limit and sum ($\lim_n \sum_{j \in S} g(j)$.
I think I "proved" something that is incorrect, but I'm not sure why it is incorrect. Clarification would be appreciated.
Suppose that $(S,F,P)$ is a probability space where $S$ is countable. Let $g:...
- 385
2
votes
1
answer
1k
views
infinite sum as integral for the counting measure
Let $u: \mathbb{N} \to \mathbb{R}^+$ be a positive sequence. Then it is true that
$$
\int u d\mu = \sum_{n=1}^\infty u(n).
$$
For the measure space $(\mathbb{N}, \mathcal{P}(\mathbb{N}),\mu)$, where ...
- 1,659
1
vote
1
answer
272
views
About James R. Munkres "Analysis on Manifolds" p.91 Theorem 11.1(b)
Definition.
Let $A$ be a subset of $\mathbb{R}^n$.
We say $A$ has measure zero in $\mathbb{R}^n$ if for very $\epsilon > 0$, there is a covering $Q_1, Q_2, \dotsc$ of $A$ by countably many ...
- 8,740
2
votes
2
answers
468
views
Showing convergence and integrability from $\sum_{n=0}^\infty \int_X |f_n| \, d \mu \lt \infty$
I have seen this question here but I didnt find a comprehensible answer so ill try my luck.
Let $(f_n)$ be a sequence of almost everywhere integrable functions defined on X with $$\sum_{n=0}^\infty \...
user439387
5
votes
1
answer
299
views
What are the values of the following integrals?
For each $k = 1,2,3,\dots, $, let $I_k=\left( \frac{1}{k+1}, \frac{1}{k}\right)$, $|I_k| = \frac{1}{k(k+1)}$. Let
$$g_k(x)=k(k+1)\chi_{I_k} \, : \, \int_0^1 g_k(x)dx=1$$
for each $k$. Let $f: \...
- 2,236