All Questions
Tagged with measure-theory integration
133 questions
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Generalizing Integration by parts for general bounded continous measure
Consider a probability measure $d\mu = w(t) dt$ with $w(t)\in L^1(I)$, $I =\left[ 0,1\right]$. What are the minimal assumption I can take on two functions $f,g:I\ \to \mathbb{R}$ so that an ...
CommunityBot
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5
votes
1
answer
914
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Extension of a function from almost everywhere to everywhere
The informal general question is: let $f$ be a "sufficiently nice" function, defined "almost everywhere". Can we develop a method to uniquely extend $f$ to the "remaining" points?
Example: Let $f(x)=\...
- 7,182
1
vote
1
answer
227
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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:
$$ \...
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1
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26
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Bivariate integration with the range of one variable shrinking to a point
Let $f(x,y)$ be a measurable function defined on $(\mathbb{R}^2, \mathcal{B}(\mathbb{R}^2))$. Define $C_{\epsilon} = \{y:d(y, y_0)\leq \epsilon\}$, then can we say for sure that the integration
$$
\...
- 53
3
votes
1
answer
199
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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 $x:K\to{\mathbb C}$, ${\...
CommunityBot
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7
votes
1
answer
621
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Does every (generalized?) Markov chain admit transition probabilities?
To pose the question let us start by recalling the following notions:
Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and $(V,\mathcal{V})$ ...
- 486
0
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1
answer
193
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$\int_{R^2}\varphi(x)d\mu(x)=0$ $\Leftrightarrow$ $\sum_{n\in \mathbb Z^2} d\mu(x-2\pi n)=0$
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 ...
- 1,018
1
vote
1
answer
206
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A boundary of the second fundamental theorem of calculus
Let's say that a set $X\subseteq [0,1]$ has Property Q if the following holds: For every continuous $f:[0,1]\to\mathbb{R}$ with $f(0)=0$ and derivative existing and bounded by 1 on $[0,1]\setminus X$, ...
- 24.2k
0
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1
answer
887
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Change of variable for integration with respect to Haar measure
I know how to estimate the integral* (see the update)
\begin{gather}
\int f(Ub)d\mu(U), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [2]
\end{gather}
where $f:S^{n-1}(\...
- 163
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0
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81
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Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?
According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that :
$$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$
where:
$X$ is separable real Banach space.
$\...
- 121
3
votes
1
answer
197
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Post composition of integral
Setup:
If $\langle \Omega, \mathfrak{F},\mu \rangle$ is a measure space, $f:\Omega \rightarrow E$ is a weakly-measurable function to a Banach space $E$, $g: E \rightarrow E'$ is a diffeomorphism and ...
CommunityBot
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4
votes
1
answer
619
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Riemannian Measures, Densities and Radon–Nikodym Theorem
If $M$ is a smooth manifold and $\mu$ is a $1$-density thereon then we may define a Borel measure (on Borel sets $A$) on $M$ as:
\begin{equation}
\nu(A) = \int_M I_A \mu.
\end{equation}
My question ...
- 34.7k
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 ...
- 11.3k
2
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1
answer
446
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Is the following "section-wise" defined function measurable in the product space?
I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form
Proposition: Assume that $(X,\...
CommunityBot
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2
votes
0
answers
181
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Is the implication ($f$ is Riemann integrable over $D_1$ and $D_2$) $\Rightarrow $ ($f$ is Riemann integrable over $D=D_1\cup D_2$) true?
Let $D_1,D_2$ be a bounded subset of $\mathbb{R}^n$ and $\partial D_1,\partial D_2$ are both of Lebesgue measure zero (that is to say: $D_1,D_2$ are Jordan measurable).
Also, let $f:D_1\cup D_2=D\...
- 35.5k
12
votes
1
answer
2k
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Naive definition of surface area doesn't work?
A first stab at a definition of surface area might go like this:
Let S be a surface. Select finitely many points from S and make a bunch of triangles having these points as vertexes. Add up the ...
CommunityBot
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7
votes
2
answers
462
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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 ...
CommunityBot
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4
votes
1
answer
819
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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-...
- 23.2k
0
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0
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454
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Reference: Bochner Integral`
What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
- 5,405
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0
answers
232
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Morphisms associated to measured spaces [duplicate]
In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...
CommunityBot
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4
votes
1
answer
740
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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 $(\cal X, \cal A)$. ...
CommunityBot
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3
votes
3
answers
2k
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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 ...
- 249
3
votes
1
answer
2k
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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 ...
- 591
1
vote
2
answers
2k
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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 $\mathrm{d}x=\mathrm{d}a\,\mathrm{d}n\,\mathrm{d}k$,...
- 2,442
5
votes
1
answer
2k
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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 ...
- 42.8k
26
votes
2
answers
12k
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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 ...
CommunityBot
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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-...
- 900
12
votes
2
answers
3k
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 ...
- 41.1k
3
votes
2
answers
1k
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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-...
- 41.1k
0
votes
0
answers
448
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 Banach ...
- 25.8k
4
votes
1
answer
670
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 $\...
- 201
7
votes
1
answer
2k
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 ...
CommunityBot
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27
votes
3
answers
5k
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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 ...
- 3,183