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27 votes
3 answers
5k views

Weak and Strong Integration of vector-valued functions

This is probably an elementary question, but outside my area of expertise, and I was unable to find any suitable reference: Suppose $f:X\to E$ is a continuous function from a compact spaces (endowed ...
Hadi's user avatar
  • 741
15 votes
3 answers
2k views

Riesz's representation theorem for non-locally compact spaces

Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For ...
Alex M.'s user avatar
  • 5,407
9 votes
2 answers
1k views

On the definition of "almost-everywhere" for non-complete measure spaces

If $(X,\mathcal{B},\mu)$ is a (non-necessarily complete) measure space, we can give two different notions of a property $P(x)$ that is true almost-everywhere : (D1) There is a measurable set $A$ ...
Jon-S's user avatar
  • 549
8 votes
1 answer
894 views

Certain compact subset of $L_1$

Let $(\Omega,\Sigma, \mu)$ be a probability measure and $X$ a Banach space. I am interested in subsets $F\subseteq L_\infty (\mu,X)$ that satisfy these two compactness conditions: $F$ is a norm-...
Rabee Tourky's user avatar
7 votes
2 answers
464 views

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace

Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
Jochen Glueck's user avatar
6 votes
1 answer
230 views

Integration along fibres of continuous map on compact Hausdorff spaces

Let $p:Z\to X$ be a continuous surjective map between compact Hausdorff spaces. Does there exist a family $m=(m_x)_{x\in X}$ of Radon probability measures on $Z$, such that the support of $m_x$ is ...
user avatar
6 votes
1 answer
223 views

Is the space of vectorial functions that are Dunford integrable complete?

Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...
Husaí Vázquez's user avatar
5 votes
2 answers
517 views

Functions whose product with every $L^1$ function is $L^1$

Let $\mu$ be a probability measure and $f$ a measurable function whose product with any integrable function is integrable: $$ \int|g|\,{\rm{d}}\mu<\infty\implies \int|fg|\,{\rm{d}}\mu<\infty. $$ ...
KhashF's user avatar
  • 3,599
5 votes
1 answer
2k views

definition of operator valued integral with spectral measure

I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011). There, they work on a Hilbert space $H$ and on the ...
Yul Otani's user avatar
  • 342
5 votes
0 answers
160 views

Hartman uniform distribution of means

Background: for a discrete abelian group $G$, a character of $G$ is a homomorphism $\chi:G\to \mathbf S^1$, $\mathbf S^1$ being the circle group $\{z\in \mathbb C:|z|=1\}$ with ordinary multiplication....
John Griesmer's user avatar
4 votes
1 answer
548 views

Two definitions of $L^p$ spaces that are not always equivalent

There are two definitions of $L^p(S, \Sigma,\mu)$ in the literature. (Here $S$ is a set, $\Sigma$ is a $\sigma$-algebra of subsets of $S$ and $\mu$ is a positive measure.) The two definitions are ...
Denis White's user avatar
4 votes
1 answer
819 views

The Notion of Strong Measurability for Separable Banach Spaces

Let $ (X,\Sigma,\mu) $ be a measure space and $ B $ a Banach space. According to my understanding, a function $ f: X \to B $ is said to be strongly $ \mu $-measurable if and only if it is the almost-...
Transcendental's user avatar
4 votes
1 answer
860 views

Lebesgue's integrability condition in several variables

The well known Lebesgue's condition of Riemann integrability says that a bounded function in one variable $f\colon [a,b] \to \mathbb{R}$ is Riemann integrable if and only if it is continuous almost ...
asv's user avatar
  • 21.8k
3 votes
1 answer
199 views

A linear functional on $C(K)^*$ continuous on each $L_1(\mu)$

I asked this at math.stackexchange, but nobody answered. Let $K$ be a (Hausdorff) compact topological space, ${\mathcal C}(K)$ the usual Banach space of continuous functions $x:K\to{\mathbb C}$, ${\...
Sergei Akbarov's user avatar
3 votes
2 answers
1k views

Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?

Hello, As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem. My question is whether each real-...
Anand's user avatar
  • 1,649
3 votes
1 answer
79 views

Closed linear span of the range of $\boldsymbol f$ and Pettis integrals of $\boldsymbol f$

Let $X$ be a noncompact locally compact topological space, let $H$ by a complex Hilbert space and let $\boldsymbol f:X\to H$ be a continuous function vanishing at infinity whose support is equal to $X$...
P. P. Tuong's user avatar
3 votes
1 answer
413 views

Schauder basis of $L^1_{\mathrm{loc}}(\mathbb{R}^n,H)$

$\newcommand{\loc}{\mathrm{loc}}$Let $(\mathbb{R}^n,\mathcal{B}(\mathbb{R}^n),\mu)$ denote the Euclidean space $\mathbb{R}^n$ with its Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ equipped with ...
ABIM's user avatar
  • 5,405
3 votes
0 answers
278 views

Radon-Nikodym derivative of vector-valued measure with respect to another vector-valued measure

Let $(X, | \cdot |)$ be a Banach space. I am interested in whether one can extend the definition of the Kullback-Leibler divergence $$ \text{KL}(\mu \ \Vert \ \nu) := \int_{\Omega} \ln\left(\frac{\...
ViktorStein's user avatar
3 votes
0 answers
101 views

Pettis vs. Dunford integrability of operator valued functions

Given a Banach space $X$ and a measure space $(\Omega ,\mu )$, one says that a function $$ f:\Omega \to X $$ is Dunford integrable, or scalarly integrable if, for every $\varphi $ in the ...
Ruy's user avatar
  • 2,263
2 votes
1 answer
239 views

Injectivity of an integral transform

For a bounded function $F: \mathbb R_{\ge 0} \to \mathbb R$ (not necessarily non-negative), is it true that $$\int_0^\infty \frac{x^ks}{(s^2+x^2)^{(k+3)/2}} F(x) dx = 0 \text{ for all $s >0$} \iff ...
Jun's user avatar
  • 303
2 votes
1 answer
425 views

Echange of Infimum Integral with Pointwise Infimum

Setup Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by $$ f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...
ABIM's user avatar
  • 5,405
2 votes
0 answers
57 views

Is the lattice of bounded Henstock Kurzweil integrable functions countably complete?

The set of HK integrable functions with an integrable upper bound $f$ forms a lattice, and satisfies the MCT and DCT. Does this mean that the lattice is countably complete? Indexing any countable set, ...
saolof's user avatar
  • 1,947
2 votes
1 answer
670 views

Integral on level sets

Let $g_\epsilon : K \subset \mathbb{R}^d \rightarrow \mathbb{R}$ (more regularity can be assumed if necessary) be defined on a compact set (with regular boundary) $K \subset \mathbb{R}^d$, and the ...
user avatar
2 votes
0 answers
162 views

$\int_{\mathbb{R}^{N}\setminus\Omega}\vert x-z\vert^{-N-\alpha} dz = c \ \forall x\in\partial U$ implies $dist(x,\partial\Omega)=c, x \in \partial U$?

Let $\alpha \in \mathbb R_+$, $\Omega \subset \mathbb R^N$ and $U \subset \Omega$. Is it true that if $$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-\alpha} dz = \text{constant} \quad \text{for all ...
user175203's user avatar
2 votes
0 answers
259 views

Bochner integral in a Fréchet space

I have a Fréchet space $V$ whose topology is (if it helps) induced by a family $\mathcal{P}$ of norms - not just seminorms - and on this space I have a Borel probability measure $\nu$. Now, I would ...
iolo's user avatar
  • 651
2 votes
0 answers
115 views

Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable

Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that $$\int_{x \in \Omega} \| u(x) \|_{\...
brighton's user avatar
1 vote
2 answers
117 views

If $f\in C([0,\infty))$, does $\delta>0$ and $g\in C^1((0,\delta))\cap C([0,\delta])$ s.t. $g\geq f$ on $[0,\delta]$ and $g(0)=f(0)$ exist?

The question is the following: Suppose $f : [0,\infty) \rightarrow \mathbb{R}$ is a continuous function. Can I find $\delta \in (0,\infty)$ and a function $g : [0,\delta] \rightarrow \mathbb{R}$ such ...
vaoy's user avatar
  • 309
1 vote
1 answer
410 views

Takesaki lemma: existence Gelfand-Pettis integral

Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras"). In another post, it was explained ...
Andromeda's user avatar
  • 175
1 vote
3 answers
580 views

Squeezing more convergence from the convergence in all $L^p$ spaces

Let $X$ be a space endowed with a finite measure $m$. Let $f_n : \to \mathbb C$ be measurable functions such that $|f_n| \le 1$ for all $n$ and $f_n \to 0$ in every space $L^p (X)$ with $p \in [1, \...
Alex M.'s user avatar
  • 5,407
1 vote
1 answer
178 views

Bochner integrability within a subspace

Let $(H,||\cdot||_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||_K$, which satisfies $$||x||_H\leq ||x||_K$$ for ...
geometricK's user avatar
  • 1,903
1 vote
1 answer
227 views

Formula for an integration on $\mathbb{Q} \cap [0,1]$

In order to work with functions defined on $\mathbb{Q} \cap [0,1]$ I would like to define an adapted "integration" formula on this set. I though that following definition could be interesting: $$ \...
Bertrand's user avatar
  • 1,199
1 vote
1 answer
89 views

Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II

This is a follow-up to this previous question, but under stronger assumptions. Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real scalar field). Let $\tilde ...
Jochen Glueck's user avatar
1 vote
1 answer
287 views

Interpolation between $L^1$ and $L^2$ spaces

I was wondering whether the following interpolation between $L^1$ and $L^2$ spaces is true: Let $f \in \mathbb{R}^n$ be such that $$ \alpha_1:= \int_{\mathbb{R}} \left\lVert f(x_1,\cdot,....\cdot) \...
Jacob Augstine's user avatar
1 vote
0 answers
87 views

$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
vaoy's user avatar
  • 309
1 vote
0 answers
54 views

Standard definition: vector-valued essential support

Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used? $$ \...
ABIM's user avatar
  • 5,405
0 votes
1 answer
193 views

$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$

Let $\mu$ be a finite measure supported by $\Gamma $ (a smoth finite curve) and absolutely continuous with respect to the length measure on $\Gamma$ such that $\Gamma \cap (\Gamma+x)$ is a finite ...
Akram Akram's user avatar
0 votes
0 answers
66 views

Reference request: Integrability condition on measures

Let $(\mathcal{C}, \|\cdot\|)$ be a (non-locally compact) Banach space with Borel $\sigma$-algebra $\mathcal{B}$. Given a probability measure $\mu : \mathcal{B}\rightarrow[0,1]$, I'm interested in ...
fsp-b's user avatar
  • 463