3
votes
3answers
72 views

Is Dirac's delta function well-defined at Lebesgue points?

Usually in textbooks, $$\int_{\mathbb{R}^d} \delta(\mathbf{x}-\mathbf{y})f(\mathbf{x}) = f(\mathbf{y})$$ holds given $f$ is continuous. On the other hand, the definition of Lebesugue point ...
1
vote
1answer
30 views

Finite Measure Space: Integral Closure = Bochner Integral

I can't sleep for so long time as the integral gives me headaches. I was looking for so many approaches. Now another one. Hope this works... Let $\Omega$ be a finite measure space and $E$ a Banach ...
0
votes
1answer
34 views

How do the inner products on $L^2$ look like?

I was wondering whether all scalar products in $(L^2[0,1],\lambda)$ are given by $\langle f,g \rangle := \int f(x)g(x) \cdot w(x) d\lambda(x)$? If this is true, what are the exact conditions that we ...
2
votes
1answer
45 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
2
votes
1answer
34 views

Spherical coordinates and Lebesgue integral

I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ...
2
votes
1answer
34 views

What is the definition of this set of absolutely continuous function

I know that $$AC(a,b):=\left\{f \in C(a,b)|f(x) = f(c)+\int_c^x g(t) d \lambda(t),c \in (a,b), g \in L^1_{\text{loc}}(a,b)\right\}$$ $$AC[a,b]:=\left\{f \in C[a,b]|f(x) = f(c)+\int_a^x g(t) d ...
0
votes
1answer
30 views

$L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
-2
votes
2answers
81 views

Group of Unitaries: Strong Continuity

Let $\mathcal{L}^2(\mathbb{R})$ be the the Hilbert space of square integrable functions, shortly $\mathcal{L}^2$. Consider the group of unitaries: $$U:\mathbb{R}\to ...
1
vote
1answer
36 views

Proposed proof Lebesgue integration question

I just want to confirm the following proof: Consider a function $u: \Omega \rightarrow \mathbb{R}$ where $\Omega \subset \mathbb{R}^{n}$ and $u \in C^{2}(\bar{\Omega})$. Let $a_{jk}$ be smooth ...
2
votes
0answers
41 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
2
votes
1answer
51 views

Simple Functions: Uniform Convergence

In the proof to proposition 4.2 of 'The Riemann Integral' it is stated that the net of simple functions converges uniformly for continuous functions. This question aims to prove this in a general ...
1
vote
0answers
64 views

Strong Notion of Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
0
votes
1answer
17 views

Lebesgue integral question using du Boise-Reymond lemma

This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in ...
1
vote
1answer
37 views

Counterexamples for L1-convergence not implying L2-convergence and vice versa

I am trying to find counterexamples for the following statements. Let $\{f_n\}$ be a sequence in $L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, and let $f$ also be in $L^1(\mathbb{R}^d) \cap ...
1
vote
0answers
27 views

Tensor Product: $\mathcal{L}^2(\Omega_1)\hat{\otimes}\mathcal{L}^2(\Omega_2)\cong\mathcal{L}^2(\Omega_1\times\Omega_2)$

I have given the realization: $$(f\otimes g)(x,y):=f(x)g(y)$$ I want to prove that the tensor product of Lebesgue spaces is isomorphic to the product Lebesgue space: ...
2
votes
1answer
62 views

Inequality for integral => Inequality for integrand

I have that for any measurable set $\Omega\subset\mathbb{R}^d$ with $|\Omega|<\infty$ \begin{align}\sqrt{\int_\Omega f(x) dx }\leq \sqrt{c\cdot|\Omega|} + \sqrt{\int_\Omega g(x) dx }.\end{align} ...
0
votes
0answers
25 views

Bounded $L^1$ functions subgroup of $L^2$

I'm currently looking at statistics and the characteristic function. And the claim is that the characteristic function must exist for every probability distribution since every probability ...
1
vote
0answers
44 views

An “academic” question on integral operators

This question is motivated by another one, asked by Cameron Williams: Adjoint of an integral operator Let us say that a Borel function $k:\mathbb R\times \mathbb R\to\mathbb C$ defines an operator on ...
2
votes
1answer
39 views

Integral result, looking for proof or reference

I am looking for a proof or reference to a text which has the following result: Let $\Omega$ be an open subset of $\mathbb{R}^{n}$. Then if $f$ is a measurable function of some sort then if it ...
1
vote
1answer
49 views

Is Lp space complete with this norm?

Let $E$ be a measurable set of finite measure and $1\leq a<b<\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space? I think this is wrong, so I tried to find a ...
2
votes
0answers
66 views

Does absolute continuity of measures imply a relation between the $L_p$ spaces?

Say $(X,\mathcal{B},\mu)$ is some measure space, and let $\sigma$ be some other measure on $(X,\mathcal{B})$ such that $\sigma\ll\mu$. What can one say about the relation between $L_p(\mu)$ and ...
2
votes
0answers
90 views

If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...
8
votes
2answers
405 views

Banach space valued integration (Riemann type)

Preface The core of any notion of integral is some sort of weighted sum: $$\sum b\mu(A)$$ Depending on wether the domain or range is decomposed these split into Riemann and Lebesgue type ones: ...
1
vote
2answers
60 views

Why is $L^p$ isomorphic to $(L^p)^2$

Is it possible to say why the spaces in the title are isomorphic as Banach spaces? Is their a Theorem that says this or is it even possible to find an explicit representation of this isomorphism?
4
votes
1answer
50 views

$L^p$-space inclusions

Let $1\leq p<q<\infty$. Which of the following inclusions are true? $L^p(0,1)\subset L^q(0,1)$ $L^q(0,1)\subset L^p(0,1)$ $L^p(0,\infty)\subset L^q(0,\infty)$ $L^q(0,\infty)\subset ...
0
votes
1answer
28 views

norm on a quotient-space

Let $M:[0,\infty)\to[0,\infty)$ be continuous and convex. Further $M$ satisfies $M(t)=0\Leftrightarrow t=0$. Let $$\mathcal L_M(\mathbb R):=\left\{f:\mathbb R\to\mathbb R \mathrm{\ measurable\ ...
2
votes
1answer
47 views

Is this space complete?

Let $X$ be the space of measurable functions $f:[0,1] \rightarrow \mathbb{R}$. I want to find out whether this space is complete under the metric $d(f,g):= \int_{[0,1]} \frac{|f-g|}{1 + |f-g|}$. Does ...
1
vote
1answer
32 views

Extend projection on $L^2$ to one on $L^p$

if we have a closed subspace of $L^p$ called $X \cong l^2$ where the topologies of $L^p$ and $L^2$ coincide (we assume $p>2$). Then we can regard $X$ as a subspace of $L^2$, which means that he is ...
1
vote
1answer
35 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
0
votes
1answer
55 views

Why is the topology of convergence in measure equivalent to this metric here?

I am currently struggeling with the topic of convergence in measure topologies. Now I read that on the space of measurable function $L^0$ on $[0,1]$ with the Borel sigma algebra and the lebesgue ...
2
votes
2answers
131 views

Biorthogonal functions in $L^p$

I asked one question that is already answered: 1.) I have a question about Lemma 9.5 on page 93/94 reference. It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't ...
1
vote
1answer
34 views

Sobolev inequality in negative index

For $s>n/2$, is it true that $$ \int |fg| dx\leq ||f||_{H^s}||g||_{H^{-s}}?$$ This inequality is used on pg 398 of the Majda Bertozzi book on Vorticity and Incompressible flow but I can't make ...
1
vote
1answer
39 views

If $u \in L^2(\Omega)$, then $\text{sign}u \in L^2(\Omega)$?

If $\Omega$ is a bounded domain and $u$ is in $L^2$, why is $\text{sign}(u) \in L^2?$ I am only stuck with the measurabilituy part. the integral is obviously finite on a bounded domain.
2
votes
1answer
31 views

Understanding the proof of completeness of $L^1$.

I'm reading the proof of completeness of $L^1 (X, \mathscr{M}, \mu)$, and I would like to clear up some confusion To prove $L^1$ is complete it suffices to show that every Cauchy sequence $(f_n)$ has ...
3
votes
1answer
55 views

Integration by parts in Bochner Lebesgue spaces.

Does there exist an analogous of integration by parts for expressions such as: $$\int_0^T {\langle u(t),v(t) \rangle }\, \mathrm{d}t,$$ where $u,v\in L^2([0,T];H)$, for some Hilbert space $H$? If so, ...
2
votes
2answers
48 views

Boundedness of a sequence of functions

Let $\Omega$ be a bounded domain and $f_n\in L^2(\Omega)$ be a sequence such that $$\int_\Omega f_nq\operatorname{dx}\leq C<\infty\qquad \text{for all}\quad q\in H^1(\Omega),\ ...
3
votes
2answers
72 views

Lebesgue integral question from wiki

I have started studying Lebesgue integration and I have a question regarding the Lebesgue integral. In the wikipedia entry of "Lebesgue integration" they define the Lebesgue integral as: Let $f: ...
2
votes
0answers
191 views

Vector-Lattices and “Approximating $\mathscr{L^1(\mathbb{R}^k)}$”.

In this question I asked whether $\mathscr{L}^1(\mathbb{R}^k)$ forms a category in any way. It was concluded that indeed it does not. I thought to myself, "well, could we at least approximate the ...
1
vote
1answer
29 views

Give examples of a basis of $L^1([0,1])$

In particular, are there any countable bases of $L^1([0,1])$?
0
votes
1answer
39 views

$\|f+g'\|_{L^2}=\|f'-g\|_{L^2}=0\Rightarrow f=g=0$ a.e?

Let $-\infty<a<b<\infty$ and $f,g\in H^1(a,b)$. So, $f,f',g,g'\in L^2(a,b)$. Suppose $$\int_a^b|f+g'|^2\mathrm dx=\int_a^b|f'-g|^2\mathrm dx=0.$$ Is it possible to conclude that $f=g=0$ ...
0
votes
2answers
44 views

Better proof for: if $u_n \to u$ in $L^2$ then $F(u_n) \to F(u)$

Let $F(v) = \int_{A} v^2(x)J(x)dx$ where $J$ is bounded. If $u_n \to u$ in $L^2$ then I want to show that $F(u_n) \to F(u)$. The proof is $$F(u_n) - F(u) = \int_A (u_n^2 - u^2)J \leq C\int_A (u_n^2 ...
0
votes
1answer
69 views

Understanding this inner product

I want to find out under which conditions on $w$, we have that $$\langle f,g \rangle :=\int_0^1 f(x)\bar{g}(x)w(x) dx $$ a dot product?, where $f,g \in C([0,1],\mathbb{C})$ and $w \in ...
2
votes
2answers
96 views

Is $C[a,b]$ a closed linear subspace of $L^{p}(a,b)$

I am not sure about the last step of my proof: $(L^{p}(X,A,\mu), \|\cdot \|)$ is a normed $L^{p}$ space of p-integrable functions. $L^{p}(a,b)$ is the space of p-integrable functions on (a,b). ...
7
votes
1answer
121 views

Dense subspace of $L^{2}[0,1]$

I know that $C[0,1]$ is dense in $L^{2}[0,1]$ but is $\{f\in C^{2}[0,1]:f(0)=f(1)=0\}$ dense in $L^{2}[0,1]$?
4
votes
1answer
39 views

For which $L^p[0,1]$, $1\le p < \infty$ does the function $n^\alpha*\chi_{[0,1/n]}$ converge weakly to 0?

My hypothesis is that this function converges to 0 weakly iff $ p < 1/\alpha$, but I am not sure how to prove this. We are working in the space $[0,1]$ with the Borel sets and Lebesgue measure. I ...
1
vote
1answer
35 views

density of spaces, Lebesgue

My question is the following: Is the space $C^\infty(\bar{G})$ dense in $L^\infty(G)$ ? Assume that $G \subset \mathbb{R}^n$. I know this result holds, if we have $G$ instead of $\bar{G}$. But is it ...
5
votes
1answer
216 views

Norm of Fredholm integral operator equals norm of its kernel?

Let $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$. Then it was fairly easy to see that $||T_k|| \le ||k||_{L^2}$, but now I was wondering how to show that ...
1
vote
1answer
29 views

approximating a measurable function using simple functions

I would like to understand the proof that shows that for any measurable function, we can approximate it with a sequence of simple functions. The proof is presented in (among others): ...
5
votes
2answers
209 views

Are integrable, essentially bounded functions in L^p?

Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$. Intuition suggests that $f \in L^p$ even for ...
1
vote
1answer
164 views

On why the Vitali Covering Lemma does not apply when the covering collection contains degenerate closed intervals

I believe I have a fundamental misunderstanding of the concept of the Vitali Covering Lemma. Definition - A closed bounded interval $[c, d]$ is said to be nondegenerate provided $c < d$. ...