Questions tagged [lebesgue-measure]
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194
questions
6
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Subset of the reals with zero inner measure and "full" outer measure in $\mathsf{ZF}+\mathsf{DC}$
Working in $\mathsf{ZF}+\mathsf{DC}$ (that is, we are allowed to use Dependent Choice but not full choice), suppose that there exists a non-measurable subset of the unit interval $[0,1]$ (just non-...
0
votes
0
answers
44
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Existence of derivative of distribution of exponential family?
Suppose $(X, \mathcal{F})$ is a measurable space and $\left\{F_\theta, \theta \in \Theta\right\}$ is a distribution family on $(X, \mathcal{F})$. When $\left\{F_\theta, \theta \in \Theta\right\}$ is ...
0
votes
0
answers
81
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A question about associated operator on continuous functions space equiped with L2 norm
$$\text{For M a connected compact manifold, T is in }C^{1+\nu}(M,M) \text{ with } \nu\in(0,1),\\ \text{i.e. DT is some Hölder continuous function with Hölder exponent }\\ \text{, Denote m as the ...
2
votes
2
answers
223
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Preimage of null sets under a monotone increasing function
Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following ...
8
votes
3
answers
564
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Distributional derivatives are locally integrable implies the distribution is also locally integrable?
Let $T$ be a distribution on $\mathbb{R}^n$ such that there are functions $f_1,\ldots,f_n \in L^1_\text{loc}(\mathbb{R}^n)$ so that $\dfrac{\partial T}{\partial x_j} = f_j, \forall j=1,\ldots,n. $
My ...
1
vote
0
answers
78
views
Finding a functional that stays non-negative on a particular subset of $L^2[0, 1]$
Start with the Hilbert space $L^2([0, 1])$ with Lebesgue measure. Fix some Borel-to-Borel measurable function $f: [0, 1] \times [0, 1] \rightarrow \mathbb{R}$. I present 4 scenarios, each more ...
0
votes
0
answers
45
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Understanding simple point processes (part 2)
This is a follow up of this previous question. I'm trying to understand the following proposition from An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods
by Daley ...
1
vote
0
answers
65
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Understanding simple point processes
Background
I'm studying the basic theory of Random Finite Sets (RFS), which is the name that is used in my field to denote simple point processes.
A simple point process is a random variable whose ...
3
votes
1
answer
236
views
Lebesgue points of a function is not affected by multiplication of the integrand with a smooth function?
Let $S^1$ be the circle, let us consider a function $f(x,t): S^1 \times [0,\infty) \to \mathbb{R}$ such that
\begin{equation}
\int_0^T \int_{S^1} \lvert f(x,t) \rvert dxdt <\infty
\end{equation}
...
1
vote
0
answers
195
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Are orbits of a measurable flow always measurable with measure zero?
Let $(X, \mathcal{B})$ be a standard Borel space with a probability measure $\mu$ on $\mathcal{B}$. Let $(T_t)_{t \in \mathbb{R}}$ be a jointly measurable flow (i.e. $(T_t)_{t \in \mathbb{R}}$ is a ...
0
votes
0
answers
120
views
Uncountable collections of sets with positive measures
Let $X$ be a compact metric space and let $T: X \rightarrow X$ be continuous. Let $\mu$ be a $T$-invariant Borel probability measure (which we can always find by the Krylov-Bogoliubov theorem).
Let $(...
2
votes
2
answers
190
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)^...
1
vote
0
answers
133
views
Analyticity of a function in two complex variables
Let $f$ be a function defined on $\mathbb{C}^2$ given by
$$
f(s,t)=\int\limits_{-\infty}^{\infty}dk_1 \int\limits_{-\infty}^{\infty}dk_2 \int\limits_{-\infty}^{\infty}dk_3 \frac{1}{\left(\sqrt{s}-k_1\...
3
votes
1
answer
134
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Recover an $L^1$ integrand by partial differentiation
Denote by $m$ the 2-dimensional Lebesgue measure on $\mathbb{R}^2$. Let $f$ be an element of $L^1(m)$ that takes only nonnegative values. Define $F : \mathbb{R}^2 \rightarrow [0,\infty)$ by
$$F(x,y) = ...
1
vote
1
answer
137
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Resources to understand Lebesgue measure of Brownian motion's path [closed]
[https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf][page 12] and [peter morters][page 47]
Let $B$ be a stanrd Brownian Motion and $R$ a function defined on $\mathbb{R}^2$ such ...
1
vote
1
answer
226
views
What is the measure of two sets which partition the reals into subsets of positive measure?
This is a follow up to this question, where I wish to partition the reals into two sets $A$ and $B$ that are dense (with positive measure) in every non-empty sub-interval $(a,b)$ of $\mathbb{R}$.
(In ...
5
votes
1
answer
356
views
Is it known that there is any function $f:\mathbb{R}\to\mathbb{R}$ at all, whose graph has positive outer measure on every rectangle in the plane?
Suppose $\lambda^{*}$ is the Lebesgue outer measure.
Question:
Does there exist an explicit $f:\mathbb{R}\to\mathbb{R}$, where:
The range of $f$ is $\mathbb{R}$
For all real $x_1,x_2,y_1,y_2$, where $...
2
votes
1
answer
147
views
Finding an explicit & bijective function that satisfies the following properties?
Suppose using the lebesgue outer measure $\lambda^{*}$, we restrict $A$ to sets measurable in the Caratheodory sense, defining the Lebesgue measure $\lambda$.
Question:
Does there exist an explicit ...
1
vote
1
answer
150
views
Any $L^\infty (\mathbb{R}^3)$ can be approximated pointwise almost everywhere by continuous function with compact support
In the book Fourier Analysis and Self-adjointess of Reed and Simon in the proof of the
Feynman-Kac formular the author states that for any $V\in L^\infty (\mathbb{R}^3)$ there is a sequence $(V_n)_n$ ...
0
votes
1
answer
78
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Convergence in sequential Lebesgue spaces
Consider a strictly increasing sequence $1\leq q_0<q_n<q_{n+1}<q$ such that $q_n\to q$ as $n\to \infty$. Let $B\subset \Bbb R^d$ be a ball, so that $L^{q}(B)\subset L^{q_{n+1}}(B)\subset L^{...
-1
votes
1
answer
57
views
The existence of a maximal “cross-sectional” filter on the Boolean algebra of measurable subsets of [0, 1] modulo almost everywhere equivalence
Let $\mathcal{B}([0, 1])$ be the Boolean algebra of measurable subsets of $[0, 1]$ modulo almost everywhere equivalence, i.e., two measurable sets which differ only by a Lebesgue null set are ...
12
votes
3
answers
677
views
If $A, B$ is a non-trivial partition of $S^1$, is it possible that $R_\theta(A) \cap B$ has measure zero for all rotations $R_\theta$?
This was previously posted to Math StackExchange. I was originally unsure whether it is suitable for posting here, but I've yet to get an answer there, so I'm just trying to see if people here can ...
8
votes
0
answers
208
views
*-homomorphisms from $L^\infty(0, 1)$ to itself that acts as the identity on continuous functions
Let $\pi: L^\infty([0, 1], \lambda) \rightarrow L^\infty([0, 1], \lambda)$ be a *-homomorphism (where $\lambda$ is the Lebesgue measure on $[0, 1]$), not necessarily normal (otherwise the question is ...
13
votes
1
answer
601
views
Almost everywhere “filling” of the continuum by the first uncountable cardinal without CH
Assuming the negation of CH, let $\omega_1$ be the first uncountable ordinal, $\mathfrak{c}$ be the cardinality of the continuum. Does there exist a map $f: \omega_1 \times [0, 1] \rightarrow \...
4
votes
0
answers
147
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Naïve definition of a measure on a fractal
This question was previously posted on MSE.
Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$.
One option would be to use ...
1
vote
1
answer
150
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Quantitative version of Lebesgue points theorem
Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \...
1
vote
0
answers
35
views
The limit set of consecutive applications of linear transforms to the single segment
Problem. Consider $n$ positive integers $1 < a_1\le \ldots \le a_n$ and $I = \left[\frac{1}{a_n - 1}, \frac{1}{a_1 - 1}\right]$. For each $a_k$ define the linear transform $\phi_k\colon x\mapsto \...
1
vote
0
answers
43
views
Measurability in a product space of a set defined only along its fibers
Consider the probability space $([0,1],\mathcal{B}([0,1]),\lambda)$, where $\mathcal{B}([0,1])$ denotes the Borel $\sigma$-algebra in $[0,1]$ and $\lambda$ is the Lebesgue measure in $[0,1]$. Then, ...
6
votes
1
answer
174
views
Concentration of volume towards the boundary
Consider a Euclidean space $X$ of large dimension $N$. For a measurable subset $G\subseteq X$ and $\varepsilon>0$ let
$$G_\varepsilon:=\{x\in G\mid B_\varepsilon(x)\subseteq G\}$$
be the set of all ...
4
votes
0
answers
281
views
If a derivative is defined everywhere and $\pm1$ almost everywhere, is it constant?
Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that the set $A:=\{x\in\mathbb{R};f'(x)\not\in\{1,-1\}\}$ has measure $0$. Does this imply that $f'$ is constant?
Context: I was ...
26
votes
2
answers
3k
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What is the origin/history of the following very short definition of the Lebesgue integral?
Typical courses on real integration spend a lot of time defining the Lebesgue measure and then spend another lot of time defining the integral with respect to a measure. This is sometimes criticized ...
3
votes
1
answer
186
views
Property of sets of positive Lebesgue measure in $\mathbb{R}^2$
Let $P\subset \mathbb{R}^2$ be a set of positive Lebesgue measure. Is it always true that a suitable rotation and translation of $P$ always contains a set of the form $\{re^{i\theta}:r\in E, \theta\...
5
votes
0
answers
147
views
Is there a natural finitely additive measure for which Vitali sets have measure zero?
Vitali sets are nonmeasurable and in particular are not null sets. But all Vitali sets are in some sense small, as described below. Let $V$ be any Vitali set and let $k \in \mathbb{N}$. For each $i \...
6
votes
0
answers
263
views
Existence of a limit of alpha-difference quotient for Hölder functions
Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that
\begin{equation}
\sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
8
votes
1
answer
2k
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How badly can the Lebesgue differentiation theorem fail?
Suppose $f:\mathbb{R}^n\to\mathbb{R}$ is integrable. Is it true that
$$
\lim_{r\to 0}\frac{\displaystyle\int_{B_r(0)}f(y)~\mathrm dy}{r^{n-1}}=0 \quad ?
$$
This is obvious if $0$ is a Lebesgue point ...
2
votes
1
answer
204
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Problem regarding set of positive Lebesgue measure in $\mathbb{R}^2$
Let $T_{p,q}$ be line joining $(0,0)$ and $(p,q).$ Now let us define the set
$$L= \bigcup_{p\in[0,1]\cap \mathbb{Q}}T_{p,1} \bigcup_{q\in[0,1]\cap \mathbb{Q}}T_{1,q} $$
and consider $P=[0,1]\times[0,...
3
votes
1
answer
293
views
$\sigma$-algebra generated by analytic sets
The Borel $\sigma$-algebra $\cal B$ on real numbers has many good properties. For instance, all continuous functions are $\cal B/\cal B$-measurable. On the other side, not only $\cal B$ is not ...
0
votes
1
answer
161
views
$f=0$ in $H^{-1}(\Omega)$ implies $f=0$ almost everywhere
Does $f=0$ in $H^{-1}(\Omega)=(H^1_0(\Omega))^*$ implies $f=0$ almost everywhere in $\Omega$?
0
votes
0
answers
46
views
Sets measurable in every affine subspace
Take a non-measurable subset $S\subseteq [-1,1]$ and subtract $S\times \{0\}$ from the unit disk $B$ in $\mathbb{R}^2$. The set $X=B\setminus (S\times \{0\})$ is measurable by 2-D Lebesgue measure ...
1
vote
1
answer
165
views
Topological analog of the Lusin-N property
$A\subset \Bbb{R}$ is meager if $A$ can be expressed as a countable union of nowhere dense sets.
Let $f:[a, b]\to \Bbb{R}$ is absolutely continuous, i.e., for every $\epsilon>0$, there exists $\...
3
votes
2
answers
506
views
Growth of $L^p$ norms as $p \to \infty$
Let $f$ be a non-negative function defined on the unit interval. It is well known that $N(p) := \left(\int_0^1 f^p(t) dt\right)^{\frac{1}{p}} $ converges to $\operatorname{esssup}_{[0,1]} f$ when $p \...
1
vote
1
answer
105
views
Measurable sets of $\mathbb R^n$ forming unique absolutely continuous convex combinations?
If we consider a finite set $A\subset\mathbb R^n$, uniqueness of the convex decomposition of points in $A$ is equivalent to the absence of $\mu\neq0$ signed measure supported on $A$ such that $\mu(\...
0
votes
1
answer
82
views
An equation in the convolution measure algebra on reals
Let us consider the measure algebra $M(\mathbb{R})$ consisting of all Radon measures on reals.
Let $\mu$ be a Radon measure in $M(\mathbb{R})$ and $\delta_0$ be the point mass measure concentrated on ...
4
votes
1
answer
191
views
Generalized limits in Boolean algebras
Let $\mathbb{B}$ be an infinite $\sigma$-complete Boolean algebra. By $\mathbb{B}^\omega$ we denote the countable product of $\mathbb{B}$ with the coordinate-wise operations. Let us call a ...
0
votes
1
answer
570
views
Is the point-wise limit of simple functions a measurable function?
Let $X$ and $Y$ be topological spaces. By a simple function $\phi: X\to Y$ we mean a finite range Borel measurable function.
Q. Is the point-wise limit of a sequence of simple functions a Borel ...
1
vote
1
answer
132
views
Isomorphy between Lebesgue space $L_1$ and the $l_1$ sum of $L_1[0,1]$ spaces
Is it true that for an infinite index set $I$, we have that $L_{1}([0,1]^{I}, \mathbb{R})$ can be written as the infinite direct sum of $L_{1}([0,1], \mathbb{R})$, i.e.
$$L_{1}([0,1]^{I}, \mathbb{R})=\...
3
votes
1
answer
295
views
Special version of Tonelli’s theorem
I am trying to prove this theorem. I have not found anything similar to it on the internet.
Special version of Tonelli’s theorem
Assume that the functions $f(x,u): [a,b] \times \mathbb{R} \to \mathbb{...
6
votes
1
answer
395
views
A problem concerning a divergent function on $[0, 1]$
This problem was posted on another forum and was given at the 1992 Miklós Schweitzer Competition. This competition is known for its very difficult problems and this one seems no exception. I also can'...
2
votes
0
answers
129
views
Weak convergence of atomic measure to Lebesgue measure
Let $G$ be the open $n$-ball in $\mathbb{R}^n$ and $G^\Delta$ the set of points in $\mathbb{R}^n$ with distance less than $\Delta>0$ from $G$.
Let $G_T=\{Tx: x\in G \}$ and $G_T^\Delta = \{ Tx: x \...
5
votes
0
answers
98
views
Which reals are Lebesgue measures of regions in $\mathbb R^n$ defined by inequalities involving polynomials with integer coefficients?
Let $a$ be a real number. What are necessary and sufficient conditions for the existence of a positive integer $n$ and a finite set of polynomials $p_1,\ldots,p_k$ with integer coefficients in $n$ ...