Tagged Questions
1
vote
1answer
24 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
0
votes
0answers
65 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) < ...
1
vote
1answer
41 views
Lebesgue integrable function and limit
Show that if $f$ is a Lebesgue integrable function on $A\subset\mathbb R$ and $$A_n=\{x\in A:|f(x)|\geq n\}$$ for $n\in\mathbb N$, then $\lim_{n\to\infty} n\cdot m(A_n)=0$.
My solution which is ...
6
votes
2answers
86 views
Calculating $\lim_{n\to\infty}\sum_{k=1}^{\infty}\frac{(-1)^{k}\arctan(n^{2}k)}{n^{\alpha}+k^{3/2}}$
I am studying for my exam in real analysis and I am having difficulties
with some of the material, I know that the following should be solved
by using the counting measure and LDCT, but I don't know ...
1
vote
1answer
111 views
$f_n$ converges pointwise to $f$ implies integral $f_n$ converges to integral $f$
Let $\lambda(E)< \infty$ and $f_n$ be measurable and continuous (on $E$) for each $n\in\mathbb{N}$.
If $f_n$ converges pointwise to $f$ (continuous on $E$) for all $x\in E$, then $\int_Ef_n ...
2
votes
2answers
70 views
Show that limit of integrals is zero
Let $\mu$ be a Borel probability measure on $(0,+\infty)$ and $\alpha > 1$. Is it true that
$$
\lim\limits_{y \to +0} y^{\alpha-1} \int\limits_0^\infty x^2 e^{-y^{\alpha} x} \, d\mu(x) = 0\ ?
...
1
vote
1answer
31 views
Limit involving probability
Let $\mu$ be any probability measure on the interval $]0,\infty[$. I think the following limit holds, but I don't manage to prove it:
$$\frac{1}{\alpha}\log\biggl(\int_0^\infty\! x^\alpha ...
0
votes
1answer
91 views
Application of Dominated Convergence Theorem.
Find $L=\lim\limits_{n \to \infty} \int_0^{n a} \exp\left(-\dfrac{t}{1+\frac{b t}{n}}\right) dt$, where $a>0$, $b>0$.
I can't see what is dominating function, but I feel that I have to use ...
1
vote
0answers
86 views
Fourier Coefficients of Complex Measure
For my homework I am trying to prove the following:
Suppose $\mu$ is a complex Borel measure on $[0,2\pi)$, and define the Fourier coefficients of $\mu$ by
$\hat{\mu}(n)=\displaystyle\int ...
1
vote
0answers
52 views
Probability measures on $\mathbb{T}$ whose Fourier coefficients tend to 1
Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent?
$\limsup_{n\to\infty}|\hat{\mu}(n)|=1$.
There exists an increasing sequence ...
5
votes
2answers
55 views
If $f_n \in L^1$, will the limit function $f$ also be in $L^1$ in monotone convergence theorem?
Monotone convergence theorem doesn't require the sequence of functions $f_n$'s to be $L^1$. When $f_n\in L^1$, will its pointwise limit function $f$ also be in $L^1$? Thanks!
2
votes
0answers
80 views
Continuity in set functions
Let a function be defined as $f:(\Omega_1,\mathcal{F}_1)\rightarrow (\Omega_2,\mathcal{F}_2)$, where $\mathcal{F}_1$ and $\mathcal{F}_2$ are $\sigma$-fields in $\Omega_1$ and $\Omega_2$ respectively.
...
1
vote
1answer
81 views
limit inferior and subsequence
I am trying to prove that if $x_n$ is a sequence such that every subsequence $a_n$ has a subsubsequence $b_n$ whose $\limsup b_n \le M $ then $\limsup x_n \le M $
if I take as a subequence $a_N = ...
2
votes
1answer
91 views
Set of measure zero?
Let $m$ be a probability measure on $W \subseteq \mathbb{R}^m$, so that $m(W)=1$.
Consider a measurable function $f:W \rightarrow \mathbb{R}_{\geq 0}$.
Say if the following holds true.
$$ \lim_{M ...
1
vote
1answer
40 views
How to understand stationary solution?
How to understand the stationary solution of the stochatic equation:
$$X_{n+1}=A_n X_n+B_n$$
And where can I find more information?
4
votes
1answer
69 views
Limit problem for $L^p$ function
I am having problems with proving the following:
Let $f$ be a $L^p$ function on $[0,1]$, $f:[0,1] \to \overline{\mathbb{R}}$. Prove that
$$\lim_{t \to \infty} t^p \mu(x: |f(x)| \geq t) = 0.$$
...
5
votes
1answer
126 views
Compute $\lim_{n\to\infty}\int_0^n \left(1+\frac{x}{2n}\right)^ne^{-x}\,dx$.
I'm trying to teach myself some analysis (I'm currently studying algebra), and I'm a bit stuck on this question. It's strange because of the $n$ appearing as a limit of integration; I want to apply ...
1
vote
2answers
81 views
$f_n$ $\in$ $L_2(\mu)$, the limit $ f \in L_2(\mu)$
If $f_n \in L_2(\mu)$, $f_n\rightarrow f$ almost everywhere, this is not enough to conclude $f\in L_1(\mu)$.
But is it enough to conclude whether $f\in L_2(\mu)$ or $$\lim_{n \to ...
2
votes
1answer
202 views
Limit Inf/Sup of Sequence of Set Example
In "A Probability Path", they have an example that states that the lim inf and lim sup of [0,n/(n+1)) is equal to [0,1). I guess I don't see how [0,1) is in all the sets except a finite number of ties ...
2
votes
4answers
117 views
Proof: Tricky limit going to 0
I'm working on a proof and to complete it I need to find a way to choose an $n$ such that $(1-a)^n < \epsilon$ for a fixed $a$ such that $\frac12 < a < 1$ and any small $\epsilon$. I'm ...
5
votes
1answer
119 views
Uniqueness of Weak Limit
As we know that weak limit of a sequence of Borel probability measures on metric space is unique. Do we have this property for general sequence of signed Borel measures on metric space? Thank you.
1
vote
3answers
131 views
The limit of integral
Let $1 \le p < \infty$ and assume $f \in L^p(\mathbb{R})$.
I'm trying to prove the limit of integral
$$\lim_{x \to \infty} \int^{x+1}_x f(t)dt =0.$$
Can I use Riesz Theorem for Banach spaces?
3
votes
2answers
121 views
Is the $ L^{p}$$[0,1]$ norm continuous in p?
I ran into the following problem when I was doing my homework, and I have no thoughts on where I should start with:
(1) If $f\in L^{2}$, show that $\displaystyle \lim_{p \rightarrow ...
1
vote
1answer
224 views
Bounded $\limsup$ integral implies $\limsup$ bounded almost everywhere?
Consider $z \in \mathbb{R}^n$ and $\{ z_i \}_{i=1}^{\infty}$ with $z_i \rightarrow z$.
Let $\phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}$. $X$ is unbounded.
I'm wondering if
$$ ...
0
votes
0answers
95 views
$\limsup$ bounded almost everywhere
Consider $z \in \mathbb{R}^n$ and a sequence $\{ z_i \}_{i=1}^{\infty}$ such that $z_i \rightarrow z$.
Let $\phi: \mathbb{R}^n \times X \rightarrow \mathbb{R}_{\geq 0}$. $X$ is unbounded.
I wonder ...
2
votes
1answer
96 views
Exercise: Limits and Probability Measure
Let $\mu$ be a probability measure on $X$ (closed but unbounded), so that $\int_X \mu(dx) = 1$.
Let functions $f_i:X \rightarrow \mathbb{R}_{\geq 0}$, $i = 1,2,...$, be Uniformly Integrable.
Prove ...
1
vote
1answer
72 views
Sequences in a Probability Measure…
Let $\mu$ be a probability measure over the (closed but unbounded) set $X \subseteq \mathbb{R}^m$: $\int_X \mu(dx) = 1$.
Consider function $f:\mathbb{R}^n \times \mathbb{R}^n \times X \rightarrow ...
0
votes
1answer
511 views
Intuition behind $\limsup$ and $\liminf$ for probabilities
I've come across these limits in Fatou's lemma, this got me massively confused.
I'd be grateful if someone could explain the intuition behind limit suprema and limit infima of probabilities (or ...
4
votes
1answer
237 views
Lebesgue integral question concerning orders of limit and integration
I've got a hand-in question in a pure analysis course that I was hoping I might get a hint on - having difficulty coming up with a decent approach.
The question:
Let $(X,\Sigma,\mu)$ be a measure ...
0
votes
1answer
256 views
Lebesgue integral calculation help
I have this limit to evaluate $$\lim_{n \rightarrow +\infty} \int_{0}^{2} \arctan \left(\frac{1}{1+x^n}\right) dx.$$
I have no idea how to solve this homework problem. Help!
0
votes
1answer
821 views
limit superior and limit inferior of the given sequence of sets
A sequence of sets is defined as $A_n=\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\}$ for some positive $p\geq0$. What is ...
5
votes
3answers
2k views
limit inferior and superior for sets vs real numbers
I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot ...
