Questions tagged [lebesgue-measure]
The lebesgue-measure tag has no usage guidance.
208 questions
4
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$\sigma$-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 ...
- 205
0
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1
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189
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$f=0$ 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
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0
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48
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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 ...
- 37.7k
1
vote
1
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183
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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 $\...
- 307
3
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2
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808
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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 \...
- 1,211
1
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1
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113
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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(\...
- 13
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1
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88
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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 ...
- 4,058
4
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1
answer
203
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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 ...
- 1,151
1
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1
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693
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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 ...
- 4,058
1
vote
1
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139
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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})=\...
- 149
3
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1
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322
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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{...
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6
votes
1
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415
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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'...
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2
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0
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137
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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 \...
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5
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0
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98
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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$ ...
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1
vote
1
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412
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Support of a measure
Let $T:X\to X$ be a continuous function on a compact manifold $X$ and let $\text{Leb}$ be the Lebesgue measure normalized so that $\text{Leb}(X)= 1$. We denote by $\mathcal{M}(X)$ the space of all ...
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1
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78
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Evaluating a limit at a discontinuity of a monotone rearrangment (distribution function)
I have a question that occurred to me and has been bothering me, because maybe graphically it seems obvious but I don't know how to get there. It has to do with the distribution function and monotone ...
- 143
2
votes
1
answer
109
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Problem regarding vanishing set of convolution
Let $f$ vanishes on an open set containing 0. So there exists $l>0$ such that $f$ vanishes on $B(0,2l).$ So we can choose $g\in C_c^\infty (\mathbb{R}^n)$ (supported on $B(0,l)$) such that $f*g$ ...
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1
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0
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78
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Cosine function evaluations are linearly independent? [closed]
Let $\{x_1,\ldots,x_n\}$ be distinct points in $\mathbb{R}^d$, and consider the functions $f_j(x) = cos(w_j^T x + b_j)$ for $w_j \in\mathbb{R}^d$, $b_j\in\mathbb{R}$, $j=1\ldots,m$, and let $m\ge n$. ...
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5
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1
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299
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Continuity of real functions
The following question concerns that without $ZF+DC$, can every function be "a little bit" continuous?
Question Is it consistent with $ZF+DC$ that for any function $f:[0,1]\to [0,1]$ and ...
- 4,201
1
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0
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100
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$ \lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)} \mathrm{d} m=0 $ associated with convergence in measure [closed]
For $m E<+\infty$, why the sufficient and necessary condition of $\left\{f_{n}(x)\right\}$ converge in measure to $0$ is
$$
\lim _{n \rightarrow \infty} \int_{E} \frac{f_{n}^{2}(x)}{1+f_{n}^{2}(x)}...
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1
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1
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136
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Prove the integral of multi-variable rational fraction is convergent
I have posted this problem in MSE long ago:
https://math.stackexchange.com/questions/3782868/multi-variable-rational-fraction-integral. But it hasn't been solved yet so I post it here. Maybe this ...
- 561
1
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0
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67
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Pontryagin's principle with Lebesgue-integrable control
Does there exist a (weak) version of Pontryagin's minimum principle in which the control is allowed to be just Lebesgue integrable? I am mostly familiar with the 1975 text of Fleming & Rishel, ...
- 1,253
2
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2
answers
875
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In practice, how is the Lebesgue measure usually generalized?
The general question
It is easy to find on the Wikipedia page for Lebesgue measure that Haar measure is a common generalization that preserves the idea of "invariance under some group action"...
- 509
2
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0
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267
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Necessary and sufficient conditions on kernels of trace-class operators
Question: Let $K \in L^2(R^n\times R^n)$. Are "explicit" necessary and sufficient conditions known such that $K$ is the kernel of some trace-class operator $A \in TC(L^2(R^n))$?
We know that ...
- 143
0
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1
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133
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Convoluted Cantor-like measure which has a continuous component [duplicate]
Let $\mu$ be a finite measure on $\mathbb R$ which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable
$$
\sum_{k\ge 1}3^{-k}X_k
$$...
- 1,352
3
votes
1
answer
166
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Coloring the uncountable Lebesgue-measurable sets of $\mathbb{R}$
A hypergraph $H=(V,E)$ consists of a set $V$ and $E\subseteq {\mathcal P}(V)$, that is, $E$ consists of subsets of $V$ of arbitrary size. Obviously, a graph is a special kind of hypergraph.
Let $H=(V,...
- 48.9k
3
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1
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479
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Almost surely convergence of translations of a measurable function
Suppose that $\alpha_n$ is a sequence of positive numbers converging to $0$.
Question. Is there a bounded measurable function $f$, say $1$-periodic, such that $f_n(x)=f(x-\alpha_n)$ does not ...
- 271
6
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1
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658
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Lebesgue–Stieltjes measure construction on $\mathbb{R}^n$
"Measure Theory and Probability Theory" by Athreya and Lahiri introduces Lebesgue–Stieltjes measure construction on $\mathbb{R}^n$ in general in the following way:
Let $f:\mathbb{R}^2\...
- 163
2
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1
answer
1k
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measure of a degenerate Gaussian distribution
I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it
in a close form.
After starting with a Gaussian random variable and restricting it to a condition, I ...
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4
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1
answer
285
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Vanishing of the product of a function and its own Fourier transform
I have found the following question to be surprisingly hard:
Is there a non-zero $f\in L^1(\mathbb R)$ or $f\in L^2(\mathbb R)$ such that
$$
f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere},
$$
...
- 1,942
0
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0
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235
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Lebesgue measure of a neighbourhood of a curve
Let $\Omega\subseteq\mathbb{R}^N$ be an open, bounded and with smooth boundary (e.g. Lipschitz boundary or more if necessary).
For any function $\phi:\Omega\to\mathbb{R},\ \phi\in C^1(\overline{\Omega}...
- 1,759
0
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1
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227
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Lebesgue measure of sets in $\mathbb{R}^N$
Let $\Omega\subseteq \mathbb{R}^N$ be an open, bounded and connected set (it can be assumed with smooth boundary if necessary).
Consider $\phi:\Omega\to\mathbb{R}$, $\phi\in C^1(\overline{\Omega})$ (...
- 1,759
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0
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88
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Sequence of open sets converge in characteristic function to an open set?
Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with Lipschitz boundary. Consider a sequence of open sets $\omega_n\subseteq\Omega,\ n\in\mathbb{N}^*$ such that there is a Lebesgue ...
- 1,759
3
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0
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774
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Simple proof of the Lebesgue density theorem in $\Bbb{R}^n$
[I posted this on MSE a while ago, but no answer was forthcoming.]
I am looking for a simple proof of the Lebesgue density theorem for $\Bbb{R}^n$. The Wikipedia page on the Lebesgue differentation ...
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1
vote
1
answer
336
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Is there a maximal translation-invariant extension of Lebesgue measure?
(Cross posted at MSE.)
The answer to this question shows that there are translation-invariant extensions of Lebesgue measure.
Are there maximal translation-invariant extensions of Lebesgue measure (...
- 869
1
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1
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160
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Summability issues of measure when we decompose a measurable set into two non-measurable parts
The question is quite "simple". Let $\lambda^*$ denote the usual Lebesgue outer measure on $\mathbb R.$ Let $E\subseteq [0,1]$ be a non-measurable subset. Do we always have
$$
\lambda^*(E) +\...
- 1,755
0
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0
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216
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Signed distance function
Let $\Omega\subseteq\mathbb{R}^N$ be an open and bounded set with uniform Lipschitz boundary. Consider the signed distance function:
$d:\mathbb{R}^N\to\mathbb{R},\ d(x)=\begin{cases} \mathrm{dist}(x,\...
- 1,759
1
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0
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63
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The Lebesgue measure of the solution space of a specific type of equation
Consider the solution space of parameter $x: D = \{x| F(x) = 0, x \in \mathbb{R}^m\}$, where $F(x) = \sum_{n=1}^{N}a_n \exp\{b_n^T x\}$ for $a_n\in \mathbb{R},b_n \in \mathbb{R}^{m}, n = 1, ..., N$.
...
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1
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Existence of integral kernel
I know the following statement ture.
Let $T \in B(L^1(\mathbb{R}^d), L^\infty(\mathbb{R}^d))$, where $B(X, Y)$ denotes all bounded linear operoters from $X$ to $Y$.
Then, $T$ has the integral kernel $...
- 193
1
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1
answer
266
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Does a subset of positive measure in $\mathbb{R}$ locally "almost" have density $1$? [closed]
Let $A\subseteq \mathbb{R}$ be a Lebesgue-measurable set. We say that $A$ is locally $\varepsilon$-dense if for any $\varepsilon > 0$, there are $x<y\in\mathbb{R}$ such that $$\frac{\mu(A\cap[x,...
- 48.9k
1
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0
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209
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The translation is continuous in $L^1(\mathbb{R}^n,d\mu)$, $d\mu=\frac{1}{1+|y|^{n+a}}dy$,$ a>0$
For any function $f\colon\mathbb{R}^n\to\mathbb{R}$, set: $\tau_hf(x):=f(x+h)$, $x,h\in\mathbb{R}^n$. Consider the following finite measure on $\mathbb{R}^n$:
$$\mu(A):=\int_A\frac{1}{1+|y|^{n+a}}\,dy$...
- 339
15
votes
2
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530
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Nontrivial signed measure on Lebesgue measurable sets being trivial on Borel sets
Let $\mathfrak{L}(\mathbb{R})$ be the collection of Lebesgue measurable sets and $\mathfrak{B}(\mathbb{R})$ be the Borel sets.
Question: Is there a nontrivial signed measure on $\mathfrak{L}(\mathbb{R}...
- 807
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1
answer
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Exchanging series and integrals
I know that I can use Lebesgue or monotone convergence theorem to exchange limit of partial sums and a Lebesgue integral, given a power series or a generic function series. But in general given a ...
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3
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296
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Shrinking subset and product
Given a segment and a value $c$ less than the segment length, let $A_1,\dots,A_n$ be finite unions of intervals on the segment. We choose a finite union of intervals $B$ with $|B|=c$ that maximizes $|...
- 1,209
5
votes
0
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160
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Extending gauge integral to higher dimensions/spaces and analogue of Riemann rearrangement theorem for it
The gauge/Henstock-Kurzweil integral allows for the integration of a very large set of functions in $\Bbb R$, at the cost of many of the nice properties of Lebesgue integration, of which it is a ...
- 151
3
votes
1
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420
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To show a set is a set of positive Lebesgue measure in $ \mathbb{R}$
Let $E\subset \mathbb{R}$ be a set of positive Lebesgue measure. Can we find $l>0$ such that $$\bigcap_{-l\leq t \leq l}t+E$$ is a set of positive Lebesgue measure?
Notation: $t+E=\{t+e|e\in E\}$
- 173
1
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0
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94
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Recursive expression of Lebesgue measure for balls with removed intersection
This is not the most theoretically challenging question; rather it is more of a reference request for a simple formula (which must be known).
Let $\left\{B_{\epsilon_n}(x_n)\right\}_{n=0}^N$ be a set ...
- 5,405
13
votes
1
answer
576
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Regarding a positive Lebesgue measure set in $\mathbb{R}^2$
Let $P\subset \mathbb{R}^2$ be a positive Lebesgue measure set. Then $P$ does not necessarily contain a subset of the form $A\times B$ where $A,B\subset \mathbb{R}$ are of positive Lebesgue measure.
...
- 173
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0
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491
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Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?
Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...
- 82.7k
5
votes
0
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140
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Measure of the boundary of an BV-extension domain: do we have $|\nabla Eu|(\partial \Omega)=0?$
Let $\Omega\subset \Bbb R^d$ be open. The space $BV(\Omega)$ consists in functions $u\in L^1(\Omega)$ with bounded variation, i.e. $|u|_{BV(\Omega) }<\infty$ where
\begin{align}\label{eq:bounded-...
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