All Questions
Tagged with measure-theory integration
133 questions
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 $x:K\to{\mathbb C}$, ${\...
- 7,426
3
votes
3
answers
2k
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 ...
- 3,793
3
votes
2
answers
994
views
measurability of integrated functions
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 homework, as ...
user11618
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-...
- 1,649
3
votes
1
answer
271
views
Expectation on a Polish space
I was wondering, if given a Polish space $X$, and given some probability measure $p$ on $X$, can the expectation of an $X$-valued function be taken? In particular, would the integral
$\int_X x dp$ ...
- 291
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 ...
user121558
3
votes
1
answer
2k
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 ...
- 591
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 $X$...
- 407
3
votes
1
answer
324
views
Does the Radon-Nikodym derivative commute with integration?
Suppose I have a measurable space $(\Omega, \Sigma)$ and a function $f: \mathbb{R} \times \mathbb{\Sigma} \rightarrow [0,1]$ such that for any $x \in \mathbb{R}$ the tuple $(\Omega, \Sigma, f(x, \_))$ ...
- 133
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 ...
- 5,405
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(...
user140442
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 ...
- 549
3
votes
1
answer
197
views
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 ...
- 5,405
3
votes
0
answers
94
views
Question on an integral inequality
I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141.
For simplicity I restae the ...
- 53
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
$$
\text{KL}(\mu \ \Vert \ \nu)
:= \int_{\Omega} \ln\left(\frac{\...
- 67
3
votes
0
answers
101
views
Pettis vs. Dunford integrability of operator valued functions
Given a Banach space $X$ and a measure space $(\Omega ,\mu )$, one says that a function
$$
f:\Omega \to X
$$
is Dunford integrable, or scalarly integrable if, for every $\varphi $ in the ...
- 2,263
3
votes
0
answers
238
views
Dominated convergence Theorem
I am struggling to understand the proof in the paper, Learning Temporal Evolution of Spatial Dependence with
Generalized Spatiotemporal Gaussian Process Models.
Theorem 2.1 in the page 33 uses ...
- 31
2
votes
2
answers
946
views
Defining definite integral using indefinite integral
Sometimes definite integral is defined using antiderivatives:
$$\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 ...
- 63
2
votes
2
answers
211
views
Limit of a integral whose integrand diverges under the limit
I am trying to simplify the following limit of integral where $\mu$ is given:
$$p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^...
- 43
2
votes
1
answer
70
views
$ \int_{E}^{*}{\psi (t) d\mu(t)}=\int_{E}{\phi (t) d\mu(t)} $
Let $(T, \mathcal{A}, \mu)$ be an arbitrary measure space.
The outer integral over $(T, \mathcal{A}, \mu)$ of a (possibly nonmeasurable) function $\psi: T\to (-\infty, +\infty]$ is defined by:
$$
\...
- 199
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_\...
- 281
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 ...
- 247
2
votes
1
answer
141
views
Injectivity of two sided Laplace transform
Let $\mu,\nu$ be finite Borel measures on $\mathbb R$.
Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide:
$$
\int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
- 769
2
votes
1
answer
239
views
Injectivity of an integral transform
For a bounded function $F: \mathbb R_{\ge 0} \to \mathbb R$ (not necessarily non-negative), is it true that
$$\int_0^\infty \frac{x^ks}{(s^2+x^2)^{(k+3)/2}} F(x) dx = 0 \text{ for all $s >0$} \iff ...
- 303
2
votes
1
answer
157
views
$\int_0^u\int_{[-1,1]^2}\int_{[-1,1]^2}\frac{1}{r}e^{-\alpha^2|x-y|^2/r} \, dx\,dy\,dr\leq Cu^{\epsilon}\alpha^{-2\beta}$
I am looking for a proof for the following fact: for $U>0,\beta>0,$ there exists $C>0,\epsilon>0$ such that $$\forall u\in [0,U],\alpha\in\left]0,1\right],\int_0^u\int_{[-1,1]^2}\int_{[-1,...
- 573
2
votes
1
answer
150
views
Duality form of $L^q$ norm, without assumption that $\int fg$ defined?
The following theorem is found, for example, in the Real Analysis books by Folland, by Yeh, and (in a slightly different form) by Royden.
Theorem. Let $(X,\mathcal{A},\mu)$ be a measure space.
Let ...
- 201
2
votes
1
answer
425
views
Echange of Infimum Integral with Pointwise Infimum
Setup
Suppose that $U$ is a subset of $L^{\infty}_{\mu}(\mathscr{F})\cap L^1_{\mu}(\mathscr{F})$ defined by
$$
f\in U \Leftrightarrow g(f(x))\leq M \mbox{ and } f \in L^{\infty}_{\mu}(\mathscr{F})\...
- 5,405
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 ...
- 91
2
votes
1
answer
446
views
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,\...
- 486
2
votes
0
answers
75
views
Have the open Questions 1 and 2 from Section 7 of the paper "Integrals with values in Banach spaces" been answered?
Some context: I had previously asked the post below on MSE, but someone suggested I ask it here and delete the original post.
In section 7 of the paper Integrals with values in Banach Spaces and ...
- 121
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, ...
- 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 ...
user140442
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
$$\int_{\mathbb R^N \setminus \Omega} |x - z|^{-N-\alpha} dz = \text{constant} \quad \text{for all ...
- 61
2
votes
0
answers
259
views
Bochner integral in a Fréchet space
I have a Fréchet space $V$ whose topology is (if it helps) induced by a family $\mathcal{P}$ of norms - not just seminorms - and on this space I have a Borel probability measure $\nu$. Now, I would ...
- 651
2
votes
0
answers
115
views
Showing that for measurable $\Omega \subseteq \mathbb{R}^n$, $L^1(\Omega; C_0(\mathbb{R}^n))$ is separable
Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that
$$\int_{x \in \Omega} \| u(x) \|_{\...
- 21
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}...
- 129
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 ...
- 5,474
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^...
- 141
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\...
- 4,597
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 ...
- 549
2
votes
0
answers
181
views
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\...
- 121
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 ...
- 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 ...
- 175
1
vote
2
answers
2k
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 $\mathrm{d}x=\mathrm{d}a\,\mathrm{d}n\,\mathrm{d}k$,...
- 486
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, \...
- 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 ...
- 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:
$$ \...
- 1,199
1
vote
1
answer
62
views
Integrability in the product space can follow from a property of the Nemytskii operator?
Let's say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Caratheodory function (i.e. $f(x,\cdot)$ is continuous for a.a. $x\in\Omega$ and $f(\cdot,t)$ is measurable for all $t\in\mathbb{R}$), where ...
- 1,759
1
vote
1
answer
190
views
Inequality and integral
Let $p(u,x):=(4 \pi u)^{-1/2}e^{-\frac{x^2}{4u}},u>0,x \in \mathbb{R}.$
Let $\mathcal{E}:=\{\phi \in C_c^\infty (\mathbb{R}),\operatorname{supp}(\phi) \subset B(0,1),\|\phi\|_\infty \leq 1\}.$
...
- 573
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 ...
- 12.6k