0
votes
0answers
17 views

Polynomial density in $L^p (\mathbb{R},\mu)$

I wanna check a necessary and sufficient condition for a Radon measure witch have the moments of all orders, to say that polynomials are dense in $L^p (\mathbb{R},\mu)$. Or just a paper or an article ...
0
votes
1answer
41 views

$\int_X f(x)\,d\mu\,\,$ exists iff $\,\,\int_X \lvert \,f(x)\rvert\,d\mu\,\,$ does

I know that, for a domain of finite measure $X$, provided that $f$ is measurable, each of the Lebesgue integrals$$\int_X f(x)d\mu\quad\text{ and }\quad\int_X |f(x)|d\mu$$exists if and only if the ...
0
votes
1answer
15 views

Measure of sum of sets of “Cauchy” sequence bounded?

Let $\{A_n\}_n$ be a sequence of sets of a $\delta$-ring $\mathfrak{M}$ of measurable sets with finite Lebesgue measure. Let us suppose that $$\forall\varepsilon>0\quad\exists ...
0
votes
1answer
27 views

Spectrum of multiplication operator by the independent variable in $L^2$

If $\mu$ is a regular Borel measure on $\mathbb{C}$ with compact support $K$, define $N_\mu$ on $L^2(\mu)$ by $N_\mu f=zf$ (the multiplication by the indipendent variable). An exercise in "Conway" ...
1
vote
3answers
38 views

Show that a metric on C[a,b] is given by $d(x,y)=\int_{a}^{b}|x(t)-y(t)|dt$

I am somewhat new to functional analysis (and this site, so please constructively chastise me if I commit any faux pas on here). I am one chapter into Kreyszig (Intro.to Func.Anal.) and I am already ...
-1
votes
0answers
54 views

Complex Measures: Lebesgue Decomposition

Disclaimer: This thread is related to: Singular Continuous Measures: Definition? Context Let $\Omega$ be a measure space with finite measure $\mu<\infty$. Consider a finite measure ...
1
vote
0answers
32 views

Fix point theorem for measures? metric on space of measures?

I have the following problem: I consider a probability space $(\Omega, \mathcal{F}, \mu)$ where $\Omega= C_0([0,1])$ (space of continuous functions on $[0,1]$ starting from 0), $\mathcal{F}$ is a ...
0
votes
0answers
26 views

About measurability for operator-valued functions

Being $E_1$ and $E_2$ Banach spaces, and working in a finite measure space, I have the following two definitions of measurability for a function $f:\Omega\to\mathcal{L}(E_1,E_2)$: $\bullet$ I say a ...
-2
votes
2answers
50 views

Complex Measures: Integrability

Approaches A complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to integrability as: $$f\in L(\mu)\iff f\in ...
1
vote
1answer
26 views

Radon-Nikodym: Integrability?

Let $\lambda:\Sigma\to\mathbb{R}_+$ and $\kappa:\Sigma\to\mathbb{R}_+$ be finite measures on $\Omega$. Then by Radon-Nikodym: $$\kappa(E)\leq L\cdot\lambda(E)\quad(\forall ...
0
votes
0answers
23 views

Extension of measure beyond Jordan-measurable sets

I know that if a set $A$ is Jordan-measurable (according to the definition that can be found here in problem 8) with respect to measure $\mu$, then, for any measure $\tilde{\mu}$ that is an extension ...
1
vote
0answers
22 views

Extension of $\sigma$-additive measure beyond Lebesgue-measurable sets.

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа an unproven statement saying that the system of sets of $\sigma$-uniqueness for a $\sigma$-additive measure $m$ defined ...
4
votes
4answers
72 views

What are non-obvious examples of measures obtained from linear functionals by the Riesz representation theorem?

In chapter two of Rudin's "Real and Complex Analysis" there is a "Riesz Representation Theorem" that dominates the chapter. My understanding of the statement of the thm. is that given a complex-valued ...
0
votes
0answers
23 views

Equality of extensions of Jordan measure

I find the following theorem in Kolmogorov-Fomin's Elements of the Theory of Functions and Functional Analysis (p. 280 of this Russian ed., p. 26 of 1963 Graylock English ed.): In order that two ...
0
votes
0answers
38 views

$\sigma$-additivity of an abstract measure

I know and have been able to prove the following lemma: Let $X$ be a set and $\mathfrak{M}$ a $\delta$-ring of subsets of $X$. The set $A\subset X$ is defined as measurable with respect to ...
4
votes
1answer
152 views

Completeness of the space of sets with distance defined by the measure of symmetric difference

Let $m$ be the measure defined on the set semiring $\mathfrak{S}_m$ and $m'$ its extension to the minimal ring $\mathfrak{R}(\mathfrak{S}_m)$. I read that $m'(A\triangle B)$ can be used as a distance ...
8
votes
1answer
98 views

Lebesgue space and weak Lebesgue space

Let $1\le p<\infty$. We define the weak Lebesgue space $wL^p(\mathbb{R}^d)$ as the set of all measurable functions $f$ on $\mathbb{R}^d$ such that \begin{equation} \|f\|_{wL^p}=\sup_{\gamma>0} ...
-3
votes
0answers
18 views

Reference: proof of Cramer-Rao

I'm looking for a detailed reference of dealing with the proof of the multivariate case of Cramer-Rao lower bound.
1
vote
1answer
34 views

Continuous linear functional and weak convergence

I have a question about a continuous linear functional. $T>0$ : fix. $C([0,T]):=\{w:[0,T]\to \mathbb{R}\,;\, w \,{\rm is\,conti.} \}$ $C_{0}([0,T]):=\{w \in C([0,T]) \,; \,w(0)=0 \}$ Then ...
2
votes
0answers
34 views

Subadditivity of Lebesgue-Stieltjes measure

Kolmogorov-Fomin's Элементы теории функций и функционального анализа define an elementary set as the finite union of intervals of the form $(a,b)$, $[a,b]$, $(a,b]$, $[a,b)$, $[a,a]:=\{a\}$ and ...
2
votes
0answers
27 views

Passing to the limit in equation with null sets (related to Bochner space)

Let $S \subset L^2(0,T;V)$ be a subset with the embedding dense, where $V$ is separable Hilbert space. Let $f:[0,T]\times V \to \mathbb{R}$ be continuous in the second argument. Suppose that for all ...
0
votes
1answer
21 views

Continuity of an integral operator

I'm stuck with this exercise: Let $A \subset \mathbb{R}$ be a measurable set. For each $f \in L^1(\mathbb{R})$ and $y \in \mathbb{R}$, let: $T(f, y) = \int_{A}{f(x-y)\mathrm{d}x}$. I have to show ...
3
votes
0answers
62 views

Equivalence of two definitions of weak solution (from a book, I don't understand something!!!!)

Consider $$y_t - \Delta y = f$$ $$y(0) = y_0$$ with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions: We have $y \in ...
1
vote
1answer
54 views

Two questions on Banach function spaces

I have recently started studying Banach function spaces over $\sigma$-finite measure spaces. By a Banach function space I mean: Let $\left(R, \mu \right)$ be a $\sigma$-finite measure space and let ...
0
votes
0answers
23 views

proof in Holders inequality,(equality) [duplicate]

I have this proof in my book: I would like to prove what I underlined in red. but I get stuck. I guess in order to get equality we only need the opposite inequality. However I still don't ...
2
votes
1answer
33 views

Polar Coordinates in $\mathbb R^n$

After proving Fubini-Tonelli theorem a formula on polar coordinates in $\mathbb R^n$ is given in my class as follows. Let $f$ be a real-valued integrable function on $\mathbb R^n$ and $S^{n-1}$ be the ...
1
vote
0answers
59 views

Complex Measures: Variations

I highly doubt a proof of mine for complex measures... Construction A complex measure can be decomposed into positive measures: ...
1
vote
1answer
48 views

Complex Measures: Integration

Disclaimer: This thread is meant as record and written in Q&A style. Additional answers are heartly welcome, too! Integration w.r.t. complex measure usually is defined via the Radon-Nikodym ...
0
votes
1answer
50 views

Spectral Measures: Integration

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. How to define the integral for unbounded measurable functions: ...
0
votes
1answer
47 views

Uniform Limit of Nets Measurable?

Clearly, the pointwise limit of a sequences of measurable functions is measurable: $$f_n\text{ measurable}\implies f\text{ measurable}\quad(f_n\to f\text{ pointwise})$$ (Especially, this holds true ...
2
votes
2answers
54 views

Spectral Measures: Support vs. Concentration

The support of a Borel spectral measure is defined by: $$\lambda\in\mathrm{supp}E:\iff E(U)>0\quad\lambda\in U\in\mathcal{T}$$ (See the german wikipedia article: Spektralmaß) Now, consider a Borel ...
0
votes
2answers
38 views

Spectral Measures: Property

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. Can you give me a hint for: $$E(A)E(B)=E(A\cap B)$$ So far for disjoints I checked: ...
1
vote
1answer
61 views

Spectral Measures: Integration of Product

Given a Hilbert space $\mathcal{H}$ and spectral a measure $E:\Sigma(\Omega)\to\mathcal{B}(\mathcal{H})$. Define the integral of simple functions by: $$\int_\Omega ...
1
vote
1answer
66 views

Bochner: Lebesgue Obsolete?

Bochner's notion of integral: $$F\text{ Bochner integrable}:\iff \exists S_n\in\mathcal{S}:\quad \int\|S_m-S_n\|\mathrm{d}\mu\to 0\quad(S_n\to F)$$ This version totally circumvents Lebesgue's notion ...
0
votes
0answers
32 views

Bochner vs. Lebesgue

I'm trying to prove that for complex functions $f:\Omega\to\mathbb{C}$ that are not a priori measurable that: $$f\text{ Bochner integrable}\iff f\text{ Lebesgue integrable}$$ Basically it reduces to ...
2
votes
2answers
78 views

Do there exist two singular measures whose convolution is absolutely continuous?

Let $\mu, \nu$ be finite complex measures with compact supports on the real line, and assume that they are singular with respect to the Lebesgue measure. Can their convolution $\mu\ast\nu$ have a ...
1
vote
1answer
32 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
0
votes
1answer
37 views

Ask for a good reference for the calculus involving singular continuous measure

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here. Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...
0
votes
1answer
27 views

Measurability of Modulus

Context: This problem came up while reading an essay on Bochner integrability. Let $\Omega$ be a measure space and $E$ a Banach space. Consider two plain functions $f:\Omega\to E$ and $g:\Omega\to ...
1
vote
2answers
76 views

The limit of $\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$ as $n\to\infty$

The task is to calculate $$\lim_{n\to\infty}\int_{0}^{1}\frac{\sqrt{n}}{1+n\ln(1+x^2)}dx$$ I tried various estimates I know to find the dominating integrable function and nothing worked. Does anyone ...
0
votes
1answer
36 views

Dirac delta distribution & integration against locally integrable function

I was reading the a lecture note online about distribution theory and it said: The Dirac delta distribution $\delta \in D'$ is defined as $\delta(\varphi)= \varphi(0) $, and there's no locally ...
1
vote
0answers
36 views

functional analysis: show L^1 integral operator has norm 1

I just started my course in functional analysis and have already stumbled across some things I don't understand, which are quite basic :(. In my lecture notes it says: Let $\mu$ be a measure on a ...
2
votes
0answers
36 views

Borel measure and positive linear forms

I'm just starting to learn about positive linear forms. If we call $C_{C}(X)$ the space of all continuous functions with compact support from domain $X$ and $\mathbb{C}$ (with $X$ a locally compact ...
2
votes
0answers
24 views

If $X$ is a LCHS and $f \in C_{C}(X)$ and $\mu$ is a Borel measure, then $f \in L^{1}(d\mu)$.

I want to prove the following statement: If $X$ is a locally compact Hausdorff topological space, and $f \in C_{C}(X)$ ($f$ is a continuous function with compact support), and if $\mu$ is a Borel ...
0
votes
2answers
52 views

Extending Positive Functionals: Linearity

How does regularity provide linearity? Given the full Banach space of bounded functions over a suitable set: $$\mathcal{B}:=\{f:\Omega\to \mathbb{C}:\|f\|_\Omega<\infty\}$$ and a linear subspace ...
4
votes
1answer
40 views

Do $\mathbb{R}^n$ and $\mathbb{C}^n$ valued ordinarily measureable functions form a Banach space under p-norm?

By measureable function I mean an "ordinarily" measureable function, that is measureable in a sense of this definition: a function between measurable spaces is said to be measurable if the preimage of ...
1
vote
0answers
58 views

Volterra operator and completely continuous operators

Consider the Volterra operator $V$ defined here. Let me give some definitions first: [Dunford-Pettis] We say that a bounded linear operator $D:L_1[0,1]\to L_1[0,1]$ is Dunford-Pettis if it sends ...
0
votes
2answers
50 views

Bochner: Absolute Integrability

For a Bochner measurable function it holds: $$f\text{ Bochner integrable}\iff\|f\|\text{ Bochner integrable}$$ for any positive measure $\lambda\geq 0$. The one direction is relatively simple when ...
2
votes
0answers
38 views

Riesz-Markov-Kakutani Theorem: Various Versions

The Riesz-Markov-Kakutani theorem usually comes in various versions. So I'm a little bit confused and wondering which of these are right. Let $\Omega$ be a locally compact space. Then: Complex ...
2
votes
3answers
116 views

Volterra operator is completely continuous

Let $\mu$ be the Lebesgue measure on $[0,1]$ on the borelians, and consider the Volterra operator $V:L^1[0,1]\to C[0,1]$ given by $$ Vf(t)=\int_0^t f d \mu $$ So, I want to show the following ...