All Questions
Tagged with measure-theory real-analysis
551 questions
2
votes
0
answers
95
views
Can we control the Wasserstein metric between $\mu$ and $\nu$ by their moment difference?
Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of ...
- 657
2
votes
2
answers
307
views
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 ...
- 113
1
vote
2
answers
271
views
An integral inequality?
Let $v \in C^\infty(\mathbb R)$ such that $1 \ge v \ge 0$ and $\int_{\mathbb R} v \, dx = 1$.
I want to show that if
$$\int_{\mathbb R} v |v''|^2 \, dx < + \infty. \tag{$\star$}$$
then
$$ \int_{\...
- 83
14
votes
3
answers
2k
views
Every function on reals a sum of two surjective real functions?
From this question, and the answer thereof, we can see that every real valued function on reals is a sum of two injective functions. Is the same true if we replace injectivity by surjectivity.
For ...
- 2,089
7
votes
2
answers
606
views
Countably representing all closed sets of positive measure
This may be a naive question, but I don't see an immediate argument.
Question: Does there exist a sequence $\{C_m\}_{m=1}^\infty$ of Borel subsets of $[0,1]$ with positive Lebesgue measure $|C_m|>0$...
- 1,959
25
votes
2
answers
2k
views
Writing a function on $\mathbb{R}$ as a sum of two injections
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. It is well-known that, using transfinite recursion with a well-ordering of $\mathbb{R}$, one can construct two injective functions $g,h: \...
- 4,265
4
votes
1
answer
340
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}
...
- 3,477
2
votes
0
answers
174
views
Product of marginals absolutely continuous with respect to a Borel probability measure
Let $\mu$ be a Borel probability measure on $\Bbb{R}^{m+n}=\Bbb{R}^m\times\Bbb{R}^n$. Consider its marginal measures $\mu_1(A):=\mu(A\times\Bbb{R}^n)\, (A\in\mathcal{B}(\Bbb{R}^m))$ and $\mu_2(B):=\mu(...
- 3,599
3
votes
2
answers
976
views
Are $L^p$ norms absolutely continuous?
Let $1 < K \leq \infty$, and suppose $f \in L^p (X)$ for all $1 \leq p \leq K$, for $X$ some $\sigma$-finite measure space with no atoms.
Question: Is the function $p \to \|f\|_{L^p}$ absolutely ...
- 6,155
-5
votes
1
answer
270
views
Calculus based on pdf [closed]
Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, ...
- 201
4
votes
2
answers
413
views
A measure assigning values in $\{0,1\}$ must be a Dirac measure?
Let $\mu$ be a measure on some measurable space $(\Omega, \mathcal F)$ such that
$$\mu(B)\in \{0,1\},\quad \forall B\in \mathcal F.$$
Can we show that $\mu$ must be a Dirac measure (under suitable ...
- 1,389
0
votes
0
answers
120
views
Mysterious Bound: $\int_{B_{4}}\|D^{2}u\|^{2} \leq 2^{n}$
I am reading through "A GEOMETRIC APPROACH TO THE CALDERON–ZYGMUND ESTIMATES" by Lihe Wang and I am perplexed by an assertion in Lemma 7. The claim is that whenever $\Delta u = f$:
$$\frac{1}...
- 1
12
votes
4
answers
1k
views
Understanding the condition $\frac{1}{p} + \frac{1}{q} = 1$ in the estimate $xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$
I just read a proof of Holder's inequality in measure theory, which boils down to the following inequality:
$$xy \le \frac{1}{p}x^p + \frac{1}{q}y^q$$
where $x,y\ge 0$ and $\frac{1}{p} + \frac{1}{q} = ...
- 7,527
9
votes
1
answer
339
views
A topological characterisation of a.e. continuity
We say a measurable function $f: \mathbb R^n \to \mathbb R$ is essentially continuous if the inverse image of any open set $O$ differs from an open set by a set of null measure, in the sense that ...
- 6,155
1
vote
1
answer
264
views
Is there a version of dominated convergence theorem for local $L^p$ spaces?
Fix $p \in [1, \infty)$. Let $(L^p (\mathbb R^d), \|\cdot\|_{L^p})$ be the Lesbesgue space of $p$-integrable real-valued functions on $\mathbb R^d$. Let $\tilde L^p (\mathbb R^d)$ be the space of ...
- 835
3
votes
3
answers
550
views
Solving interval problems without outer measure
Is it possible to solve the following two problems on intervals using elementary methods, without using the outer measure ?
Problem 1
If $(I_n)$ is a disjoint sequence of subintervals of interval $I$ ...
0
votes
2
answers
125
views
Is there a modification of $f$ on a null set such that $F: [0, T] \to L^p ({\mathbb R}^d), t \mapsto f(t,\cdot)$ is Bochner measurable?
Let $T>0$ and $p \in [1, \infty)$. Let $f \in L^p ([0, T] \times {\mathbb R}^d)$. By a theorem in this thread, there is a Lebesgue null subset $N$ of $[0, T]$ such that $f(t, \cdot)$ is Lebesgue ...
- 835
1
vote
1
answer
258
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 ...
- 63
10
votes
1
answer
1k
views
Within ZFC, is $2^{\aleph_0}<2^{\aleph_1}$ provable/independent?
So, I ask whether from the ZFC axioms one can prove X that every uncountable set has strictly more than continuum many subsets, or whether X is independent of the ZFC axioms. Note that (within ZFC) ...
- 3,584
5
votes
1
answer
366
views
Quantitative Lebesgue density theorem
Let $A \subset [0, 1]$ be a measurable set, and $\mathbf 1_A$ its indicator function, viewed as a function on $\mathbb R$. Define for each $\delta > 0$, the function $f_{A, \varepsilon}: \mathbb R \...
- 6,155
5
votes
1
answer
222
views
If every point is a Lebesgue point of $f$, does $f$ satisfy the intermediate value property?
Let $f: \mathbb R \to \mathbb R$ be a locally integrable measurable function.
We say $f$ satisfies the intermediate value property if given any $a, b\in \mathbb R$ with $a < b$, whenever $u \in \...
- 6,155
4
votes
1
answer
836
views
Can a function that is continuous on a dense set be almost extended to a continuous function?
Note: All sets and functions defined below are assumed measurable. $\mu$ denotes the Lebesgue measure.
Let $D$ be a dense subset of $[0, 1]$, and $f: D \to \mathbb R$ a function. Given $\varepsilon &...
- 6,155
3
votes
1
answer
162
views
If $f : [0,1] \to H$ has $t$-derivative with respect to the norm of $H$, and $H=L^2[0,1]$ itself, does the $t$-derivative exist in ordinary sense?
The question is as in the title.
Let $H$ be a separable Hilbert space and $f : [0,1] \to H$ be a continuous mapping such that
\begin{equation}
f'(t):=\lim\limits_{\alpha \to 0} \frac{f(t+\alpha)-f(t)}{...
- 3,477
9
votes
1
answer
764
views
Does the family of fat Cantor sets contain a measurable rectangle?
Let $S \subset (0, \frac{1}{3}) \times [0, 1]$, be the set such that for each $0 < t < \frac{1}{3}$, $S \cap (\{ t \} \times [0, 1])$ is the standard Smith-Volterra Cantor set of parameter $t$.
...
- 6,155
2
votes
0
answers
175
views
Banach space of vector measures
Let $S$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $S$. Let $A$ be a Banach space over the field of complex numbers. A countably additive map $\mu:\Sigma\to A$ is called a vector ...
- 552
8
votes
1
answer
687
views
Measure without measurable sets
This question is a little on the softer and speculative side, so bear with me.
Usually a measurable space is $(\Omega, \Sigma)$, a set $\Omega$ and sigma algebra $\Sigma$ of subsets. A measurable ...
- 3,574
1
vote
0
answers
99
views
Proving more stronger fomula for discrepancy of a sequence [closed]
I am reading famous book about uniform distribution of sequences by Kuipers and Niederreiter and have questions about solving below exercise from that book. Before going to main exercise I will write ...
- 111
0
votes
1
answer
245
views
Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e
The Riemann-Liouville integral is defined by
$$
I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t
$$
where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
- 141
2
votes
1
answer
77
views
Total sets for $L^p$ for every $1\leq p < \infty$
Consider $L^p[ 0,1]$ for $1\leq p < \infty$ or, if you prefer, $L^p(\mu)$ where $\mu$ is a finite Borel measure with compact support. Let $(\phi)_{i\in I}$ be a subset of measurable functions that ...
- 355
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:
$$
\int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
- 759
3
votes
0
answers
94
views
Question on an integral inequality
I am reading van de Vaart and Weller, Weak Convergence and Empirical Processes With Applications to Statistics. And I am stuck in the proof of Theorem 2.6.7 on page 141.
For simplicity I restae the ...
- 53
3
votes
1
answer
100
views
Vague Topologies induced by $C_c$ and $C_0$ are the same on a closed ball of finite Radon measures?
Let $X$ be a locally compact Hausdorff space. Denote $C_c(X)$ and $C_0(X)$ the space of continuous functions with compact support and vanishing at infinity respectively. By Riesz representation ...
- 315
0
votes
1
answer
74
views
$\int_{\mathbb{R}}|p(v-r,x)-p(u-r,x)|\,dx \leq C\frac{v-u}{u-r}$
Consider $p(u,x)=(4\pi u)^{-d/2}e^{-\frac{|x|^2}{4u}},u>0,x\in \mathbb{R}^d.$
Prove that there exists $C>0$ such that for all $0<u\leq v,r\in[0,u[,$ $$\int_{\mathbb{R}^d}|p(v-r,x)-p(u-r,x)|\, ...
- 573
0
votes
1
answer
165
views
An inequality involving the essential supremum
Let $\mu$ be a Radon measure on $[0, 1]$, and $f: [0, 1] \to \mathbb R$ a Borel measurable function.
Question: Is it true that for $\mu$ almost every $x \in [0, 1]$, we have
$$f(x) \leq \mu\text{-...
- 6,155
2
votes
1
answer
157
views
$\int_0^u\int_{[-1,1]^2}\int_{[-1,1]^2}\frac{1}{r}e^{-\alpha^2|x-y|^2/r} \, dx\,dy\,dr\leq Cu^{\epsilon}\alpha^{-2\beta}$
I am looking for a proof for the following fact: for $U>0,\beta>0,$ there exists $C>0,\epsilon>0$ such that $$\forall u\in [0,U],\alpha\in\left]0,1\right],\int_0^u\int_{[-1,1]^2}\int_{[-1,...
- 573
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
0
votes
1
answer
112
views
Integral and inequality
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
0
votes
1
answer
248
views
Integral with inequality
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
4
votes
2
answers
391
views
Lebesgue differentiation theorem at boundary points for Sobolev traces
$\newcommand{\R}{\mathbb R}$
Let $\Omega\subset \R^d$ be a smooth, bounded open set and fix $p\geq 1$.
Fact 1: the usual Lebesgue differentiation theorem says that, if $u\in L^p(\Omega)$, then
$$
u(x)...
- 5,380
2
votes
1
answer
215
views
Measuring how "close" $\alpha\in[0,1]\setminus\mathbb{Q}$ is to being rational
Let $\mathbb{N}_+$ denote the set of positive integers and let $\mathbb{N}_0 = \mathbb{N}_+\cup\{0\}$. Fix $\alpha\in[0,1]\setminus \mathbb{Q}$. For $n\in\mathbb{N}_+$ we let the approximation radius ...
- 48.9k
1
vote
2
answers
107
views
Measurability of Brjuno numbers
A positive irrational number $\alpha\in{\mathbb R}\setminus {\mathbb Q}$ is said to be a Brjuno number if $$\sum_{i=1}^\infty\frac{\log q_{i+1}}{q_i} < \infty$$ where $q_i>0$ is the denominator ...
- 48.9k
0
votes
1
answer
230
views
Can we further restrict the space of test functions to $C_c^\infty (X)$ in weak convergence?
Let
$X := \mathbb R^n$,
$C_b(X)$ the space of all real-valued bounded continuous,
$C_c(X)$ the space of all real-valued continuous functions with compact supports, and
$C_c^\infty(X)$ the space of ...
- 657
6
votes
1
answer
287
views
A characterisation of continuous real functions
Let $f: \mathbb R^n \to \mathbb R$ be a measurable function.
We say $f$ is precise if for every $x \in \mathbb R^n$ and every compact subset $K$ of $\mathbb R^n$ such that for $|K \cap B_\delta (x)|&...
- 6,155
2
votes
0
answers
116
views
A technical question concerning convolution product
Let $v\in L^p(\Bbb R^d)$, $1\leq p<\infty$ be nonzero function, i.e., $v\not\equiv 0$.
Define $$u(x)= |v|*\phi(x)= \int_{\Bbb R^d} |v(y)|\phi(x-y)d y$$ with $\phi(x)= ce^{-|x|^2}$ and $c>0$ so ...
- 1,101
4
votes
1
answer
668
views
Optimal Transport: how is this transport map Borel measurable?
I'm reading Theorem 1.17. and its proof at page 14 of Santambrogio's Optimal transport for applied mathematicians. The content is not hard but a little bit long (because of related detail). Please ...
- 835
3
votes
1
answer
236
views
Property of sets of positive Lebesgue measure in $\mathbb{R}^2$
Let $P\subset \mathbb{R}^2$ be a set of positive Lebesgue measure. Is it always true that a suitable rotation and translation of $P$ always contains a set of the form $\{re^{i\theta}:r\in E, \theta\...
- 73
4
votes
1
answer
134
views
On partial absolute continuity
$\newcommand\B{\mathscr B}\newcommand\A{\mathscr A}\newcommand\si{\sigma}$Let $I:=[0,1]$, and let $\B$ and $\B^2$ denote the Borel $\si$-algebras over $I$ and $I^2$, respectively. Let $\A$ stand for ...
- 128k
1
vote
0
answers
64
views
Prescribed class of measurable sets
Let $X\neq\emptyset$ and let $\mu:P(X)\to[0,\infty]$ be an outer measure. Recall that, a set $A\subseteq X$ is $\mu$-measurable if
$$
\mu(B)=\mu(A\cap B)+\mu(B\setminus A), \text{ for all }B\subseteq ...
- 895
5
votes
0
answers
163
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 \...
- 303
0
votes
1
answer
327
views
Deduce that a function is zero on interval $[0,M]$
I have been thinking about this for the last few days but I was not able to produce a definitive answer.
Take an integrable function $g$ that maps in $\mathbb{R}$ and with domain contained in $[0,M]$ (...
- 141