All Questions
115
questions
3
votes
1
answer
67
views
Prove that $\int _Xf_ngd\mu \overset{n\to\infty}{\to}\int _Xfgd\mu$ ,$\forall g\in \mathcal{L}^\infty (\mu )$ if it's true $\forall g\in C_b(X)$
Let $X$ be a Polish space and $\mu :\mathfrak{B}_X\to\overline{\mathbb{R}}$ a finite measure on the Borel subsets of $X$. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of $\mathcal{L}^1(\mu )$ and $f\...
- 1,209
4
votes
1
answer
61
views
Finding a function which is $L^1$ not $L^2$ and the integral is bounded by the square root.
So I have been trying to solve the following this past exam problem:
Find $f\in L^1(\mathbb{R})$, not $L^2(\mathbb{R})$ with the property:
$$
\int_{A}|f(x)|dm\leq \sqrt{m(A)}\quad\text{ for all } A\...
- 394
0
votes
0
answers
43
views
Is it possible to define $L^p$ spaces using a non-sigma-finite measure space and a Banach space?
Most often (at least in probability), one defines the $L^p$ space as
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $p\geq 1$ be a real number. Then
$$
L^p(\Omega, \mathcal{F},...
- 5,247
5
votes
1
answer
78
views
Is the function $x \mapsto \mu(A+x)$ continuous, where $\mu$ is a finite Borel measure on $\mathbb R^n$ and $A \in \mathcal B(\mathbb R^n)$
Let $\mu$ be a finite regular Borel measure on $\mathbb R^n$ and $A$ is a Borel set. I am trying to prove that $x \mapsto \mu(A+x)$ is continuous. Here $\mu$ is regular means it satisfies assumptions ...
2
votes
1
answer
110
views
Let $(X, \mathcal M, \mu)$ be a finite measure space. If $f_n, f$ are $L^1$ functions, $f_n$ unif. integrable, $f_n \to f$, then $f_n \to f$ in $L^1$
Let $(X, \mathcal M, \mu)$ be a measure space with $\mu(X)$ finite, and $f,f_1,f_2,\dots$ be $L^1$ functions. Show that if $\{ f_n \}$ is uniformly integrable and $f_n \to f$ for a.e. $x \in X$, then $...
- 3,800
4
votes
1
answer
184
views
Counting measure and integrability condition
Let $\mathcal{A}$ be the $\sigma$-field on $[0,1]$ that consists of all subsets of $[0,1]$ that are (at most) countable or whose complement is (at most) countable. Let $\mu$ be the counting measure on ...
- 103
2
votes
1
answer
101
views
Separability of $L^p(\mathbb R^n ,\mu)$
Let $\mu$ be a Radon measure in $\mathbb R^n$ it is true that $L^p(\mathbb R^n,\mu)$ is separable?
I not find a proof of these fact, I only would know that these is true.
- 332
2
votes
1
answer
78
views
Proving $\|fg\|_{L^1} \leq \|f\|_{L^p}^{\alpha} \|g\|_{L^q}^{1-\alpha}$
I am trying to prove the following inequality:
$$\|fg\|_{L^1} \leq \|f\|_{L^p}^{\alpha}\|g\|_{L^q}^{1-\alpha}$$
where
$$\quad 1= \frac1{p} +\frac1{q}, \quad 1 \leq p, q, \leq \infty.$$
In particular, ...
- 6,277
0
votes
0
answers
44
views
A question on maximal operators on $\mathbb{R}^2$
Let $f\in L^1(\mathbb{R}^d),$ let $\widetilde{M}f$ be the unrestricted maximal function
$$\widetilde{Mf}(x_0, y_0) = \sup_{Q}\frac{1}{|Q|}\int_Q |f(x,y)|\,dx\,dy. $$
where the supremum is over all $Q=...
- 5,611
1
vote
0
answers
61
views
An inequality relating to maximal functions and convolutions
For $f\in L^1(\mathbb{R})$, let us denote the restricted maximal function $Mf$ as
$$(Mf)(x)=\sup_{0<t<1}\frac{1}{2t}\int_{x-t}^{x+t}|f(z)|\,dz.$$
I would like to show that
$$M(f*g)\le M(f)*M(g)$$...
- 5,611
2
votes
2
answers
88
views
If $\Vert f \Vert_p < \infty$ for some $1 < p < \infty$ prove that $\lim_{t\to \infty} t^{-(1-1/p)} \int_0^t f(x) \, dx =0$.
Question: Suppose $f \colon [0,\infty) \to \mathbb{R}$ is measurable so that $\Vert f \Vert_p < \infty$ for some $1 < p < \infty$. Prove that
$$\lim_{t\to \infty} t^{-(1-1/p)} \int_0^t f(x) \,...
- 707
2
votes
1
answer
71
views
Limits using layer-cake representations for $L^p$ space involving two functions
Question: Let $f,g \colon \mathbb{R} \to \mathbb{R}$ be such that $g \in L_{loc}^1(\mathbb{R})$ and $f$ is measurable. If $\int_\mathbb{R} |f|^p |g| \, dx < \infty, 0 < p < \infty$ show that
$...
- 707
0
votes
1
answer
65
views
Bochner integration without completing measures?
Consider a positively measured space $(S,\Sigma,\mu)$ and a real or complex Banach space $X$. Is it possible to build the Bochner integral and the Bochner spaces $L^p(S,\Sigma,\mu;X)$ without ...
- 2,007
2
votes
1
answer
85
views
Prove that $\int_{\mathbb{R}^3} f(x, y)g(y, z)h(z, x) dλ_3(x, y, z) ≤ ∥f∥_{L_2(\mathbb{R}^2 )}∥g∥_{L_2(\mathbb{R}^2 )}∥h∥_{L_2(\mathbb{R}^2 )}$
Hey I have this problem where I am stuck on solving it.
I Think it is very easy but I dont know how to proceed.
The Exercise is
Let $f, g, h ∈ L_2(\mathbb{R}^2 )$.
To show is:
$\int_{\mathbb{R}^3} f(x,...
- 900
1
vote
0
answers
49
views
Hardy's Integral Inequality for $\alpha > -1$ for nonnegative functions with $p \in [1,\infty)$
I want to show that that for $d\mu(x) = x^\alpha dx, d\nu = x^{\alpha + p} dx$
We have
$$\int_0^\infty \left(\int_x^\infty f(t)dt\right)^p d\mu(x) \leq c \int_0^\infty f^p(x) d\nu(x) \quad \alpha < ...
- 39