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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 ...
14 votes
1 answer
655 views

Almost all non-negative real numbers have only finitely many multiples lying in a measurable set with finite measure

Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that Lebesgue measure of $A$ is positive i.e. $0<\lambda(A)<\infty$. Let $S$ be the set defined as follows: $$S:=\{t\in [0,\infty):nt\...
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) \|_{\...
14 votes
1 answer
919 views

Was Cantor aware of Lebesgue theory of integration?

Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twentieth century. Lebesgue theory has a deep connection with ...
0 votes
1 answer
55 views

Looking for a family of random variables such that only the second clause is fulfilled [closed]

Working with the epsilon-delta-criterium, a family $(X_i)_{i \in I}$ on $(\Omega,A,P)$ is uniformly integrable if i) $sup_{i \in I} E(X_i) <\infty$ ii) $\forall \epsilon>0$ ex. $\delta>0$ s.t....
14 votes
5 answers
13k views

The Fundamental Theorem of Calculus in Lebesgue Theory

I am interested to what extent the famous identity $$ \int_a^b f'(x) \ dx=f(b)-f(a) $$ is true for a function $f:[a,b]\to \mathbb C$ continuous on $[a,b]$ and differentiable on $(a,b)$. One famous ...
6 votes
1 answer
343 views

Is there a standard way of defining the integral of an extended real function with respect to a finitely additive probability measure?

Let $X$ be a set, and let $\mu$ be a finitely additive probability measure defined on $2^X$. Let $\Phi$ be the set of functions from $X$ to $\mathbb R \cup \{-\infty, \infty\}$. Is there a standard ...
2 votes
0 answers
84 views

How to define Lebesgue Integrability of functions assuming values in an arbitrary topological vector space over an arbitrary topological field?

I have already asked this question in this MSE thread, but some people suggested me to ask to the MO community also. Preliminaries An algebra of sets in a set $X$ is an $\mathcal{X}\subseteq\mathcal{P}...
5 votes
1 answer
435 views

Integration theory for functions and values with values in topological rings

I am curious whether somebody ever tried to generalize the classical theory of Lebesgue integral to functions and measures with values in Hausdorff topological rings. The generalization of a measure ...
5 votes
1 answer
319 views

Spherical average of $\frac{1}{x}$

Let $X_1,...,X_n$ be points on $\mathbb S^1.$ We then define the expectation value $E(X)=\frac{1}{n}\sum_{i=1}^n X_i.$ Let $\frac{dS(X_1)}{2\pi}$ be the normalized surface measure of $\mathbb S^1,$ i....
9 votes
1 answer
557 views

Deep applications of the Pettis integral?

In the Notes section of chapter 2 of Diestel and Uhl's Vector Measures they make the comment: "Presently the Pettis integral has very few applications. But our prediction is that when (and if) ...
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 ...
2 votes
1 answer
70 views

$ \int_{E}^{*}{\psi (t) d\mu(t)}=\int_{E}{\phi (t) d\mu(t)} $

Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space. The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by: $$ \...
5 votes
1 answer
426 views

When is the Radon-Nikodym derivative locally essentially bounded

Let $\mu\lll\nu$ be $\sigma$-finite Borel measures, which are not finite, on a topological space $X$. Under what conditions is $0<\operatorname{ess-supp}(\frac{d\mu}{d\nu}I_K)<\infty$ for every ...
0 votes
1 answer
153 views

$ \{X_n\mathbb{1}_{X_n\in[-n,n]}\}$ is uniformly integrable

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space. Suppose $\{X_n\}$ is a sequence of random variables satisfying : $$ \sup_{n}{\mathbb{E}(|X_n|)} <\infty $$ Suppose that $$ \dfrac{M_j}{...
2 votes
1 answer
150 views

Duality form of $L^q$ norm, without assumption that $\int fg$ defined?

The following theorem is found, for example, in the Real Analysis books by Folland, by Yeh, and (in a slightly different form) by Royden. Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space. Let ...
1 vote
1 answer
101 views

Integral average near a point of dispersion

Let $\Omega\subset\subset\mathbb R^{n}$ be a bounded domain and let $E\subset \Omega$ be a Lebesgue measurable set. Let $f\in L^{1}(\Omega)$ and let $x\in \Omega$ be a point of dispersion of $E$, that ...
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? $$ \...
2 votes
0 answers
200 views

The collection of mean value abscissas in the Mean value theorem

The integral mean value theorem for continuous f on [0,b] and finite positive continuous measure $\mu$ we have $$\frac{1}{\mu[a,b]}\int_{a}^{b}f(x)d\mu(x)=f(c)(*)$$ for at least one $c\in [a,b]$. We ...
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 ...
4 votes
0 answers
136 views

Integration on a family of differential forms

Let $X$ be a smooth manifold, and denote by $\Omega^*(X)$ the set of all smooth differential forms on $X$. Assume we have a family of differential forms $\omega_t \in \Omega^*(X)$, $t\in E$, ...
3 votes
0 answers
238 views

Dominated convergence Theorem

I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with Generalized Spatiotemporal Gaussian Process Models. Theorem 2.1 in the page 33 uses ...
1 vote
0 answers
106 views

Change variable in integration with symmetry

Not sure if I can ask such fundamental problem here. Let $G$ be a finite group, $\sigma \in G$. Consider linear group actions fo $G$ on $\mathbb{R}^n$. $\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(...
2 votes
1 answer
498 views

Uniform sampling on a Riemannian manifold via tangent space and exponential map

Given a Riemannian manifold $(\mathcal{M}, \{g_x\}_{x \in \mathcal{M}})$ and a fixed point $x \in \mathcal{M}$, does the following procedure yield uniform samples from $\{y \in \mathcal{M} : d_\...
5 votes
0 answers
202 views

Invariant measure on coset space and integrable functions

Let $G$ be a locally compact abelian group, and $H$ a closed subgroup. Let $C_c(G)$ be the space of continuous, compactly supported complex valued functions on $G$. General theory of Haar measure ...
3 votes
2 answers
235 views

A reduction problem from $\mathbb{R}^2$ to $\mathbb{R}$

Let $f,g \in L^1_\text{loc}(\mathbb{R})$, with $g \geq 0$, and such that for almost every $(x,y) \in \mathbb{R}^2$, at least one of the following equations is true : \begin{align*} f(x) + f(y) + g(...
2 votes
0 answers
1k views

Definition of the surface measure in some books

I am studying PDEs and in some books (Folland, Introduction to Partial Differential Equations and Evans, Partial Differential Equations), I found an integral integrated by the surface measure on a $C^...
19 votes
1 answer
1k views

Fubini without CH

In Real and Complex Analysis, Rudin gives an example (due to Sierpinski) of a function $f:[0,1]^2\to[0,1]$ separately Lebesgue-measurable in each argument, such that $$ \int_0^1 dx\int_0^1f(x,y)\,dy \...
2 votes
0 answers
263 views

Is a maximal set of rectangles known for which Lebesgue’s Differentiation Theorem holds true?

Lebesgue's differentiation theorem states that if $x$ is a point in $\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a Lebesgue integrable function, then the limit of $\frac{\int_B f d\...
1 vote
0 answers
145 views

How to show that this function is continuous (Geometric Measure Theory)

I want to prove that the function $F: \mathbb{R}_+ \to \mathbb{R}$ defined by $$F(t)=\int_{\{d=t\}} g \, d\mathcal{H}^{n-1}$$ is continuous if $g:\Omega \subset \mathbb{R}^n\to \mathbb{R}$ is ...
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})\...
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 ...
3 votes
1 answer
2k views

How "compact" are sets of finite measure?

Let $K$ be a compact set of $\mathbb R^n$, then every open cover of $K$ will have a finite subcover. Now consider the following situation: Everything I say in the following is with respect to the ...
2 votes
1 answer
2k views

Explicitly representing a random variable in terms of indicator functions

Motivation: I want to compute $$E[g(X)] := \int_{\Omega} g(X(\omega)) d\mathbb{P}(\omega) \tag{*}$$ without needing change of variable formula. I want to prove the change of variable formula (you ...
0 votes
1 answer
139 views

Change of variables for double integral [closed]

Thank you for your time. My basic question is whether the following change of variables allowed $$\int_0^a \int_0^b f(a-b)g(b-c)h(c)\,dc\,db = \int_0^a \int_0^b f(c)g(b-c)h(a-b)\,dc\,db$$ I fail to ...
2 votes
2 answers
946 views

Defining definite integral using indefinite integral

Sometimes definite integral is defined using antiderivatives: $$\int_{a}^b{f(t)dt}=F(b)-F(a)$$ where $F$ is any continuous function such that: $$(\forall t\in[a,b]\setminus C)(F'(t)\text{ exists and ...
32 votes
3 answers
3k views

What are the obstructions for a Henstock-Kurzweil integral in more than one dimension?

I have recently come across the book The Kurzweil-Henstock Integral and its Differentials by Solomon Leader, in which, if I understand correctly, the HK integration process is modified in a way that ...
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 ...
4 votes
2 answers
1k views

Regularity of measures in the theorem of Riesz

There are two concurrent theories of measure/integration on a locally compact topological spaces: either as positive linear forms on the space of continuous functions with compact support, or as Borel ...
2 votes
1 answer
242 views

Conditions for a monotonic integral average

I am looking for conditions that ensure that an integral average of a function from $\mathbb R^n$ to $\mathbb R$ is a monotonic function of the averaging set. To be more specific, let me start with ...
1 vote
0 answers
340 views

Integrating a function with respect to a mixture measure

This builds off on an old question about mixture measures: Generalized notions of mixture Suppose $\mathcal{M}$ is a family of probability measures, and $Q$ is a probability measure over $\mathcal{M}$...
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) \...
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 $A$ ...
2 votes
0 answers
478 views

Does equality almost everywhere on a product imply equality almost everywhere on sections [closed]

(This question was on MSE, with no answers) Consider two $\sigma$-finite measure spaces $X_1$ and $X_2$, and $X=X_1\times X_2$ the product measure space (a priori non-completed). Take two functions ...
9 votes
3 answers
3k views

What is the Dunford Integral and why is it useful?

Wikipedia http://en.wikipedia.org/wiki/Pettis_integral defines the Pettis Integral for Banach space valued functions wrt to some measure space by duality. It calls the Pettis & Bochner integral ...
10 votes
2 answers
1k views

Can the integration of integrable sections of a measurable function of two variables ever result in a non-measurable function?

I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct ...
3 votes
1 answer
938 views

Stokes theorem for manifolds with boundary as disjoint union of submanifolds

Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my ...
0 votes
1 answer
376 views

How to prove the equality on the Fourier transformation of measure? [closed]

I cannot prove the following equality on the Fourier transformation of measure: let $\mu$ be a finite Borel measure on $R^d,$ then $$\lim\limits_{T\to \infty}\frac{1}{(2T)^d}\int_{[-T,T]^d}|\widehat{\...
0 votes
1 answer
179 views

Theory of integration of Kernel in çinlar probability and stochastic

I'm reading the probabilistic book write by çinlar, but I don't understand the Kernel theory, in details: $ (E,\mathcal{E}),(F,\mathcal{F})$ are two measurable space $$K:E \times \mathcal{F} \...
4 votes
1 answer
968 views

Usable Change-of-Variables Formula for Hausdorff Measure

Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula: $$ \int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...