All Questions
93
questions
1
vote
2
answers
90
views
Find functions fitting conditions
Give an example of a sequence of functions $\{u_n\}_{n \geq 1}$ that are positively measureable on $(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mu)$ that fulfills $u_n \rightarrow u$, for $n \rightarrow \...
- 11
0
votes
1
answer
109
views
$\lim_{n\to\infty}\int_X f_n d\mu=2023$ and measures
Let $(X,\mathcal{A},\mu)$ be a measure space and let $\{f_n\}_{n\geq1}$ be a sequence in $\mathcal{M}_{\bar{\mathbb{R}}}^+(\mathcal{A})$. Assume that $f_n\to f$ for $n\to \infty$ and that
$$\lim_{n\to\...
user1197800
2
votes
0
answers
78
views
Prove an Integral Function is Differentiable
Everything is under the Lebesgue integral setting. Fix $p \in \mathbf{R}$. Define for $y \in \mathbf{R}$,
$$
F(y) := \int_0 ^\infty \frac{\sin(xy)}{1 + x^p} \,dx.
$$
It can be proven that $F(y)$ is ...
- 1,196
3
votes
1
answer
56
views
Is this proof of monotone convergence theorem circular?
I'm reading below result from Amann's Analysis III. The Google book link to the actual page is here.
3.4 Theorem (monotone convergence) Suppose $\left(f_{j}\right)$ is an increasing sequence in $\...
- 17.6k
1
vote
0
answers
126
views
Let $f \in L^1 (\mathbb{R}), f_n(x):=f(x-\frac{1}{n})$. Prove that $f_n$ converges to $f$ in $L^1(\mathbb{R})$. [duplicate]
I'm trying to solve some real analysis question, and I have no clue how to solve this one.
$$ \text{Let: }f \in L^1 (\mathbb{R}), f_n(x):=f(x-\frac{1}{n})$$
$$
\text{Prove that } f_n \text{ converges ...
- 31
0
votes
2
answers
63
views
convergence and integral: a small question [closed]
For any $t>0$ suppose that $f_t$ is a continuous function on $\mathbb{R}$ and uniformly bounded in $t$ : $\|f_t\|_\infty \leq C$. Suppose that for any $g \in L^1(\mathbb{R})$ we have
$$
\int_\...
- 323
0
votes
1
answer
45
views
Is this function of class $L^{1}(\mathbb{R})$, $L^{2}(\mathbb{R})$, both or none?
I have the following function:
$$\frac{\sin(x)}{x^{3/2}}$$
To prove it is $L^{1}(\mathbb{R})$, one has to prove this integral converges:
$$\int_{-\infty}^{+\infty} \frac{|\sin(x)|}{|x|^{3/2}}\, dx$$
...
0
votes
2
answers
52
views
A "convergence theorem" in measure theory?
Let $A_n$ ($n \in \mathbb{N}$) be a sequence of measurable subsets of some measure space with $A_1 \subseteq A_2 \subseteq...$ and let $A:= \bigcup_{n < \infty} A_n$. Let $f:A \to \mathbb{R}$ a ...
- 2,436
1
vote
1
answer
169
views
Monotone Convergence Theorem applied to the limit of a real-valued function
I have a given function $f(x,\omega)\geq 0$, where
$x\in\mathbb{R}$ is a parameter and $\omega\in\Omega$, some sampling
space. $f(x,\omega)$ is increasing in $x$ for any $\omega$. I am interested in ...
- 1,027
1
vote
1
answer
50
views
Show that the operator $L_M$ has a convergent subsequence for any fixed M.
Let $a,b \in \mathbb{R}$ and ${\{{u_n}}\}_{n\in\mathbb{N}}$ bounded in $X:=L^P\left([a,b]\right)$ with $1 \leq p < \infty$ such that:
$\forall\, \varepsilon>0,\,\exists\,\,\delta>0$ such that ...
- 101
2
votes
1
answer
41
views
Dominated convergence theorem with function $\frac{1}{x^{\frac12+\frac1n}}\left(\sin\frac{\pi}{x}\right)^n$
For $n\geqslant 3$ and $x\in(0,\infty)$, we define $$f_n(x)=\frac{1}{x^{\frac12+\frac1n}}\left(\sin\frac{\pi}{x}\right)^n.$$
I have to calculate $$\lim_{n\to\infty}\int_{[1,\infty)} f_n(x)\,dx.$$
...
- 2,420
4
votes
1
answer
254
views
Convergence in measure metrizable?
I am trying to show for a $\sigma$-finite measure space $f_n\rightarrow f$ $\mu$-stochastically iff $\lim_{n\rightarrow\infty} d(f_n,f)=0$ where
$$d(f,g):=\sum^\infty_{k=1}\frac{2^{-k}}{\mu(\Omega_k)}\...
1
vote
1
answer
51
views
Limits Under the Integral Sign
Let $$F_n:\mathbb R \to \mathbb R$$ be a sequence of positive measurable functions s.t. $F_n \to F$ pointwise and
$$\int_{\mathbb R} F_n\ d\lambda \to \int_{\mathbb R} F\ d\lambda <\infty$$
let $...
- 1,199
0
votes
1
answer
140
views
Convergence of the integral of positive part sequence of measurable function
Let $( X, \Sigma , \mu)$ be a measure Space
and $(f_n)$ be a sequence of Borel measurable
functions. Suppose that $f _ { n } \rightarrow f$ Point wise and $\lim _ { n \rightarrow \infty } \int f _ { n ...
- 2,184
0
votes
2
answers
134
views
Show that : $ \mathbb {E}(|X|)\leq \liminf_{n\rightarrow\infty}(\mathbb {E}(|X_n|)) $
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $\{X_n\}_{n\in \mathbb{N}}$ is a sequence of random variables in $L^1$.which converges to $X$ in probability and a constant $M>0 $ ...
- 39