All Questions
Tagged with lp-spaces measure-theory
1,365
questions
4
votes
1
answer
81
views
weak convergence and pointwise implies $L_p$ convergence
Suppose $f_i \to f$ weakly in $L^p(X, M, \mu)$, $1 < p < \infty$, and that $f_i \to f$ pointwise $\mu$-a.e. Prove that $f_i^+ \to f^+$ and $f_i^- \to f^-$ weakly in $L^p$.
My proof:
Since $f^\pm ...
- 1,575
0
votes
1
answer
71
views
How do we know the dual pairing between Lp spaces is well defined? [closed]
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space and let $X \in L^p(\Omega, \mathcal{A}, \mu)$ and $Y\in L^q(\Omega, \mathcal{A}, \mu)$. Then the dual pair betweent these spaces is defined as $\...
- 365
1
vote
2
answers
235
views
Is there a smooth function, which is in $L^1$, but not in$L^2$? [closed]
I am studying measure theory. While going over $L^p$-spaces I asked myself, whether there is $f\in C^\infty(\mathbb{R})$ s.t. $f\in L^1(\mathbb{R})\setminus L^2(\mathbb{R})$? I assume there could be ...
- 29
1
vote
1
answer
47
views
What justifies the use of global coordinates when computing the $L^p(\mathbb{T}^n)$ norm?
Consider the $n$ dimensional torus $\mathbb{T}^n$. The $L^p$ spaces over $\mathbb{T}^n$ is defined as consisting of an equivalence class of functions satisfying:
$$\int_{\mathbb{T}^n}|f|^p < \infty....
- 6,275
1
vote
0
answers
17
views
Does the sequence of bounded symmetric square integrable holomorphic functions have a convergent subsequence?
Let $f$ be a bounded holomorphic function on $\mathbb D^2$ and $s : \mathbb C^2 \longrightarrow \mathbb C^2$ be the symmetrization map given by $s(z) = (z_1 + z_2, z_1 z_2),$ for $z = (z_1, z_2) \in \...
- 2,612
1
vote
0
answers
36
views
Is the set of $L^2([0,1])$ functions $g$ s.t. $g\circ \psi = f\circ \phi$ for fixed $f\in L^2$ and some $\phi_*(dx)=dx, \psi_*(dx)=dx$ closed?
Consider the $L^2$ space for the Lebesgue measure $dx$, i.e., the set of functions $f:[0,1]\to \mathbb{R}$ such that $\int_{0}^{1}|f(x)|^2dx<\infty$. Fix one function $f\in L^2$ and the space of $...
- 1,565
0
votes
1
answer
34
views
$L_p$ inequality for measurable sets
Let $(U,\mu)$ a finite and positive measure space, and $1\leq p<\infty$. Suppose that for every $\varepsilon$ and measurable subset $A\subset U$, there exists a measurable subset $B\subset U$ such ...
3
votes
1
answer
167
views
A problem in L1 space
Problem: Let $(X, \mathcal{A}, \mu)$ be a measure space. Let $f: X \to [0, \infty)$ be measurable. Then define the set $$A_f = \left\{g \in L^1 (\mu)\ |\ |g| \leq f\mbox{ a.e.} \right\}.$$
Prove the ...
- 596
0
votes
1
answer
31
views
A Multiplication operator in a Hilbert space: $M_h$ is bounded and $||M_h|| \leq || h||_{\infty}$ [duplicate]
I'm trying to understand the example below, taken from Axler's Measure Integration and Real Analysis book.
How does one prove that $M_h$ is bounded and that $||M_h|| \leq || h||_{\infty}$?
I was ...
- 5,331
2
votes
1
answer
37
views
Proof: If $\mu$ is $\sigma$-finite and $\mathscr{A}$ is countably generated, then $L^p(X,\mathscr{A},\mu)$ $1\leq p<+\infty$ is separable.
Background
I have some trouble understanding a step of the proof of the following proposition:
Proposition$\quad$ Let $X$ be a measure space, and let $p$ satisfy $1\leq p<+\infty$. If $\mu$ is $\...
- 2,493
2
votes
1
answer
38
views
Proving that convergence of norms and convergence a.e. implies strong convergence
I have in my notes the following theorem
Theorem
$(Y,\mathcal{F},\mu)$ $\sigma-$finite measure space, $p\geqslant 1$, $\{f_n\}\subset L^p(Y)$ sequence of functions, $f\in L^p(Y)$ such that $$\lim_{n\...
- 516
2
votes
1
answer
50
views
$L_p$ norm estimate of a sum
Let $R>0$, and $(B_k)_{k \in \Bbb{N}}$ a collection of disjoint balls of radius R and let $f$ be a measurable function on $\mathbb{R}^n$ of the form
$$f = \sum_{k=1}^{\infty} a_k X_{B_k}$$
for ...
- 22k
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
6
votes
2
answers
96
views
Understanding the proof of $L^p(X,\mathscr{A},\mu)$ is complete ($1\leq p<+\infty$)
Background
I have some questions when reading the proof of $L^p(X,\mathscr{A},\mu)$ is complete for $1\leq p<+\infty$. The proof is proceeded by showing that each absolutely convergent series in $L^...
- 2,493
1
vote
1
answer
30
views
Given $f \in L^p_{\text{loc}}(\Omega) \setminus L^\infty(\Omega)$, does it follow that $A \cap S(f,K) \neq \emptyset$ for all $K > 0$?
Context. Throughout this post I will be dealing with the Lebesgue measure over $\mathbb R^n$. Moreover, I denote the measure of a measurable set $E \subset \mathbb R^n$ by $|E|$ and $\Omega \subset \...
- 1,141