All Questions
209
questions
0
votes
1
answer
59
views
Estimating integrals and measures over Hilbert space using finite dimensional projections
Let $H$ be a separable Hilbert space with orthonormal basis $\{e_k\}_{k \in \mathbb{N}}$. Let $P_n$ be the projection onto the $n$-dimensional subspace of $H$:
$$P_n x = \sum_{i=1}^n \langle x, e_i\...
- 6,277
4
votes
1
answer
55
views
If $g_1, g_2\in\mathscr{L}^{\infty}(X,\mathscr{A},\mu)$ are equal locally $\mu$-almost everywhere, then $T_{g_1}=T_{g_2}$.
Background
Suppose that $(X,\mathscr{A},\mu)$ is an arbitrary measure space, that $p$ satisfies $1\leq p<+\infty$, and that $q$ is defined by $\frac{1}{p}+\frac{1}{q}=1$. Let $g$ belong to $\...
- 2,493
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
1
vote
1
answer
45
views
Weak convergence in $L^2$ equivalence
Problem statement: Denoting by $B_r$ the open ball of $\mathbb{R}^N$ centered at the origin with radius $r$, consider a sequence $f_n \in L^2(B_1)$ which is bounded in the $L^2$ norm. Prove that $f_n$ ...
- 608
0
votes
0
answers
67
views
Some questions about integration and operator theory.
As we all know there are multiple integral operators which all basically do the same thing in various contexts. I am talking about operators like the Lebegues integral, Riemann integral and more ...
- 103
2
votes
0
answers
39
views
Isometry between the dual space $L^1(\mu)^*$ and $L^\infty(\mu)$. In need of a 'trial function'.
I am currently trying to show that $L^p(\mu)^*$ is isometrically isomorphic to $L^{p'}(\mu)$, where p' is the dual exponent. We are working in a $\sigma$-finite measure-space $(X,\mathcal{E}, \mu)$.
I ...
1
vote
1
answer
95
views
How can I prove that $\Vert \int _Xfd\mu \Vert \leq \int _X\Vert f\Vert d\mu $?
Let $(X,\Sigma,\mu )$ be measure space. Denote by $\mathbb{R}^{m\times n}$ the set of the matrix $m\times n$. Suppose that $f:X\to \mathbb{R}^{m\times n}$ is a function whose coordinates are ...
- 1,209
3
votes
0
answers
87
views
Two formulations of Riesz–Markov–Kakutani representation theorem
Let $ X $ be a locally compact Hausdorff space, and $ C_c(X) $ be the space of all complex-valued continuous functions with compact support on $ X $.
As far as I know, there are two formulations of ...
- 884
0
votes
1
answer
31
views
Non-zero $\mathrm{BV}$ function integrating to zero
Let $I=[0,1]$ and $\mathrm{BV}(I)$ be the space of all functions $f:I\to I$ with
$$||\cdot||_{\mathrm{BV}(I)}=\mathrm{var}_I(\cdot)+||\cdot||_{L^1(I)}.$$
I found a claim in a few papers that any 2-...
- 1,136
2
votes
1
answer
94
views
Finding dual operator - applying Fubinis theorem
For X=Y=$C([0,1])$ find the dual operator $T^*$ of $Tf(t)=\int_0^tf(s)ds$
Using the Riesz representation theorem we get $$\int_0^1f(s)d\nu(s)=T_\mu^*(f)=\int_0^1(\int_0^tf(s)ds)\;d\mu(t).$$
Now I'd ...
- 143
5
votes
1
answer
78
views
Is the function $x \mapsto \mu(A+x)$ continuous, where $\mu$ is a finite Borel measure on $\mathbb R^n$ and $A \in \mathcal B(\mathbb R^n)$
Let $\mu$ be a finite regular Borel measure on $\mathbb R^n$ and $A$ is a Borel set. I am trying to prove that $x \mapsto \mu(A+x)$ is continuous. Here $\mu$ is regular means it satisfies assumptions ...
4
votes
1
answer
184
views
Counting measure and integrability condition
Let $\mathcal{A}$ be the $\sigma$-field on $[0,1]$ that consists of all subsets of $[0,1]$ that are (at most) countable or whose complement is (at most) countable. Let $\mu$ be the counting measure on ...
- 103
2
votes
1
answer
78
views
Proving $\|fg\|_{L^1} \leq \|f\|_{L^p}^{\alpha} \|g\|_{L^q}^{1-\alpha}$
I am trying to prove the following inequality:
$$\|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
3
votes
2
answers
195
views
On a locally-compact group $h(x^{-1})$ constant a.e. $\implies h(x)$ constant a.e. (proof of uniqueness of Haar measure)
I'm trying to understand the final step in a proof of uniqueness of left Haar measures on locally-compact groups as presented in Knapp's "Advanced Real Analysis" (pg. 224-225) or this note ...
- 1,434
1
vote
1
answer
59
views
A proof of the Riesz-Markov for compact subsets of the complex numbers.
I had the following idea of how to give a simple proof of the Riesz-Markov-Kakutani representation theorem for subsets of the complex numbers and I wanted to know whether someone sees some principle ...
- 137