3
votes
0answers
70 views

Non Borel Spaces: Gauge Integral

Question Is there a generalization of the gauge integral to measure spaces that do not necessarily arise out of some topology? I'm wondering since it seems as the gauge crucially uses ...
3
votes
2answers
326 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 ...
3
votes
1answer
307 views

Integral wrt probability measure

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
7
votes
2answers
245 views

Integration on Compact Semirings

I want to know if integration of functions $f:X\rightarrow G$ where $G$ is a compact semiring has been defined and if it is possible to ensure that all continuous functions are integrable. This is ...
1
vote
1answer
253 views

From Lebesgue Integral to Stieltjes Integral, and integration by parts

Let $X$ be a real random variable with c.d.f function $F$. Let $g$ be an increasing measurable real function and assume that $\mathbb{E}\left[g(X)\right]$ exists (and is finite). What additional ...
1
vote
1answer
147 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 ...
2
votes
2answers
352 views

How do these two Haar measures on SL(2,R) compare?

By using the Iwasawa decomposition, one obtains a (bi-invariant) Haar measure on $G:=\mathrm{SL}(2,\mathbb{R})$ which can be symbolically written as ...
1
vote
2answers
282 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) \space exists ...
3
votes
1answer
508 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 ...
6
votes
2answers
2k views

About the definition of Borel and Radon measures

I am trying to understand the notion of Radon measure, but I am a little bit lost with the different conventions used in the litterature. More precisely, I have a doubt about the very definition of ...
8
votes
1answer
419 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 ...
9
votes
2answers
643 views

Borel sets preserved under open maps?

Given open map f: $R^n$ to $R^n$ such that each open set $U\in R^n$, $f(U)$ is also open. Are Borel sets in $R^n$ preserved under f? Motivation: Pre-image of Borel sets under continuous map is a ...
2
votes
3answers
356 views

Defining the integral of a function using the product measure

Imagine that we're trying to define the expression $$\int_U f(x)dx$$ in a rigorous way. Assume that $f:X \rightarrow \mathbb{R}^{\geq 0}$ where $(X,\mu)$ is a measure space, and suppose that $U$ is a ...
10
votes
3answers
2k views

Counterexamples to differentiation under integral sign?

I'm exploring differentiation under the integral sign (I want to be much faster and more assured in doing this common task). So one thing I'm interested in is good counterexamples, where both ...
5
votes
3answers
646 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 ...
2
votes
2answers
440 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 ...
0
votes
0answers
344 views

Is an integrable map from a measure space to a Banach space always measurable?

Is every integrable mapping defined in a general measure space to a Banach space measurable? The answer is yes if it is function (real valued). The answer is yes if it is a mapping into a ...
7
votes
1answer
799 views

Universally measurable sets and weak topology

After I posted this question, a couple of months ago, and got from MO-users several good hints, I think i'm ready, after some study, to ask another related question (or rather, to focus on the main ...
2
votes
2answers
714 views

measurability of integrated functions

Hello everybody, DISCLAIMER: I'm not a mathematician, but a computer scientist, so I hope the question is not trivial (or perhaps I hope so, in order to get a definitive answer). Anyway it's not a ...
4
votes
1answer
389 views

Is the 2d gauge integral equivalent to the Lebesgue integral for nonnegative functions?

Let $f$ be a function from $[0,1]\times [0,1]$ to $\mathbb{R}$. Definition: 2dgauge$\displaystyle\int f \; = \; I$ $\Leftrightarrow$ For all neighborhoods $U$ of $I$, there exists a function ...
16
votes
3answers
2k 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 ...
19
votes
2answers
1k 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 ...
48
votes
8answers
6k views

Why is Lebesgue integration taught using positive and negative parts of functions?

Background: When I first took measure theory/integration, I was bothered by the idea that the integral of a real-valued function w.r.t. a measure was defined first for nonnegative functions and only ...
1
vote
2answers
562 views

Defined Almost Everywhere

How can one prove that the convolution of f \in L^1 and g \in L^p is defined almost everywhere? Here f and g are measurable functions in R^n. In general what techniques are there for showing some ...