# Tagged Questions

**10**

votes

**1**answer

232 views

### smooth Luzin theorem

For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with ...

**-2**

votes

**1**answer

103 views

### a question regarding the interchange the order of finite summation with finite integration [closed]

Question (1) What are the conditions the complex function $f_n(t)$ and real parameter $B>1$ and positive integer $N>1$ need to satisfy such that the interchange of the finite summation with ...

**0**

votes

**0**answers

58 views

### Does there exists $f\in A_{\mathbb R}(\mathbb T)$ with $||f||=r$ such that $||e^{if}||=e^{r}$?

Let $\mathbb Z$, the set of integers, be a group with respect to addition and its dual group is the $\mathbb T = \{z\in \mathbb C : |z|=1\}$ , one dimensional torus. Put,
$\ell^{1}(\mathbb Z)= ...

**1**

vote

**0**answers

81 views

### Does the difference quotient of an absolut cont. funct. converge in L^1?

Assume that $\mu$ is a finite Radon measure on the real line and $f$ is integrable wrt. $\mu$. Define
$F(x)=\int_{]\infty;t]}f(y)d\mu(y) $
Is the following statement true?
The functions ...

**0**

votes

**0**answers

60 views

### Continuity of the real Monge Ampère operator on convex functions

Let $E\subset\mathbb{R}^n$ be a convex set, $u:E\to\mathbb{R}$ a convex function and $B\subset E$ a Borel set.
We define the (multivalued) gradient
\begin{array}{rccl}
\nabla [u]:& \mathbb{R}^n ...

**4**

votes

**0**answers

191 views

### decreasing rearrangements: why the asymmetry of measure-preserving maps?

Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is ...

**1**

vote

**2**answers

184 views

### An almost orthogonality principle for L^p

I recently asked this question on Math StackExchange and someone suggested that it would probably be more suited for Math Overflow. Since it still has not been answered, here it goes:
If two ...

**0**

votes

**1**answer

186 views

### Variation on Fatou's lemma for Sobolev norms

Recall that Fatou's Lemma says that for every sequence $f_n$ of non-negative measurable functions
$$\int \liminf_{n\to \infty} f_n \ d\mu\leq \liminf_{n\to \infty} \int f_n\ d\mu \ .$$
If I am not ...

**1**

vote

**1**answer

189 views

### Injective with Finite Discontinuities Mapping from $\mathbb R^n$ to $[0,1]$

Hi,
as a continuation to the fully answered question:
Injective&Intregrable Mapping from $\mathbb R^3$ to $\mathbb R$
Can one think of an injective $f:\mathbb R^n\rightarrow[0,1]$ that has only ...

**3**

votes

**2**answers

353 views

### Injective and Integrable Mapping from $\mathbb R^3$ to $\mathbb R$

Is there an injective and Riemann integrable map $f:\mathbb R^3\rightarrow\mathbb R$? (Of course such a map cannot be continuous.)

**5**

votes

**2**answers

261 views

### Measures of full Hausdorff dimension for self-affine sets

Consider the iterated function system $T_{1}(x)=(\beta x,\tau y)$, $T_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set ...

**1**

vote

**1**answer

139 views

### Uniform equicontinuity of a family of indefinite integrals

Let $f_k$ be a sequence of measurable functions on $\mathbb{R}^k$ where $k > 1$. (Let us be generous and also assume that $f_k$ is locally integrable.) Does anyone know what the phrase
uniform ...

**2**

votes

**1**answer

276 views

### Atoms of regular Borel measure

Let $X$ be locally compact Hausdorff space. Let $\mu$ be a Borel measure on it which is finite on compact and outer regular with respect to open sets and inner regular with respect to compact sets. ...

**0**

votes

**1**answer

257 views

### Why are simple functions defined for positive coefficients (in measure theory) [duplicate]

Possible Duplicate:
Why is Lebesgue integration taught using positive and negative parts of functions?
Hey,
I am currently referring 'probability with martingales'. To develop lesbegue ...

**2**

votes

**1**answer

665 views

### A question about measurable structures on function spaces

Hey, I was just wondering, I'm using some of Robert Aumann's ideas about measurable structures on function spaces (From his paper 'Borel structures for Function spaces': ...

**0**

votes

**1**answer

765 views

### Functionals continuous with respect to weak convergence

It's well known that a functional of the form $u \mapsto \int f(u) dx$ is continuous with respect to weak convergence (say weak* convergence in $L^\infty$) if and only if the function $f$ is affine. ...

**3**

votes

**0**answers

257 views

### For METRIZABLE spaces, do the Banach classes and Baire classes coincide?

In this paper: 'Borel structures for Function spaces' by Robert Aumann,
http://projecteuclid.org/euclid.ijm/1255631584
Aumann claims that when X and Y are metric spaces (among other things), the ...

**4**

votes

**0**answers

700 views

### Exceptional Set in Egoroff's Theorem

I'll use the version of this question I posted on Stakexchange to replace the former version. To simplify the situaton, we assume the measure space we are dealing with right now is a closed interval ...

**10**

votes

**3**answers

1k views

### Infinitesimal generators of stochastic processes

What's the $L^1$ analogue of Stone's theorem saying that any strongly continuous 1-parameter unitary groups has a unique self-adjoint generator?
More precisely: let $X$ be a measure space ...

**2**

votes

**2**answers

292 views

### Existence of rearrangements of functions in $L^p([0,1])$ when given a measure preserving map

Given a funtion $f \in L^p([0,1])$ (take $p=\infty$ if you'd like), and also a measure preserving map $s:[0,1] \to [0,1]$ (meaning $s$ pushes Lebesgue measure forward to itself) I would like to know ...

**4**

votes

**1**answer

367 views

### Jordan measurability of the level sets

Let $A$ be a compact subset of $R^n$ and $d_S(\bullet, A)$ be the
signed distance function of $A$. Namely, $d_S(p,A) =
d\left({p,\partial A} \right)$ for p in A, and $d_S(p,A) =
-d\left({p,\partial ...

**3**

votes

**4**answers

825 views

### Amenable exponential growth

Dear forum members,
Does anyone have a clear example of an amenable group with exponential growth?
Is real that if G is virtually amenable (has an amenable subgroup of finite index) then it is ...

**8**

votes

**0**answers

465 views

### G-delta of measure 0 containig the rationals.

It is well known that $\mathbb{Q}$ is not a $G_\delta$. In fact no countable dense subset of $\mathbb{R}$ is a $G_\delta$. We order the rationals in a sequence: $\mathbb{Q}=\lbrace r_k\rbrace$, and ...

**47**

votes

**23**answers

15k views

### Why should one still teach Riemann integration?

In the introduction to chapter VIII of DieudonnÃ©'s Foundations of Modern Analysis (Volume 1 of his 13-volume Treatise on Analysis), he makes the following argument:
Finally, the reader will ...

**10**

votes

**1**answer

671 views

### Who first found this characterization of Lebesgue integration?

Write $L^1$ for the Banach space $L^1([0, 1])$. Given $f \in L^1$, define $f_1, f_2 \in L^1$ by
$$
f_1(x) = f(x/2),
\qquad
f_2(x) = f((x + 1)/2).
$$
Let $I = \int_0^1$. Then $I$ is the unique ...

**18**

votes

**2**answers

3k views

### Is there a category structure one can place on measure spaces so that category-theoretic products exist?

The usual category of measure spaces consists of objects $(X, \mathcal{B}_X, \mu_X)$, where $X$ is a space, $\mathcal{B}_X$ is a $\sigma$-algebra on $X$, and $\mu_X$ is a measure on $X$, and measure ...

**18**

votes

**1**answer

641 views

### A collection of intervals that can cover any measure zero set

This is a follow-up to this question (in fact, this is what originally motivated me to ask that one.)
Let's say that a sequence $\{s_i\}$ of positive reals covers a set $X\subset\mathbb R$ if there ...

**16**

votes

**3**answers

2k views

### 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 ...

**6**

votes

**2**answers

550 views

### How small can the set of $p$ such that the $L^p$ norms are different for two fixed functions?

What does it tell you about two functions if their $L^p$ norms are the same for all $p\in[1,\infty]$? Certainly they could be related by composition with a diffeomorphism with Jacobian of norm 1, or ...

**12**

votes

**11**answers

2k views

### Applications of Measure, Integration and Banach Spaces to Combinatorics

I'm going to be teaching a Master's level analysis course(measure theory, Lebesgue integration, Banach and Hilbert spaces, and if there's time, some spectral or PDE stuff) in the fall. My problem is ...

**6**

votes

**4**answers

903 views

### Measure 0 sets on the line with Hausdorff dimension 1

I use $\dim_H(E)$ to denote the Hausdorff dimension of a set $E \subseteq \mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if ...

**5**

votes

**2**answers

1k views

### Discontinuous convolutions

Is the following true?
The convolution of two infinitely differentiable as well as integrable real functions can be nowhere continuous.
A reference/proof idea would be very helpful.

**0**

votes

**0**answers

223 views

### Density of Dolean exponentials in L2 and Wiener Measure

Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by ...

**13**

votes

**4**answers

1k views

### Is Conway's base-13 function measurable?

Robin Chapman introduced me to Conway's Base 13 Function. Now, my real analysis is a tiny bit rusty, so maybe my question has a really simple and quick answer, but here it goes:
Consider the support ...

**8**

votes

**2**answers

2k views

### Why pi-systems and Dynkin/lambda systems? On the relative merits of approaches in measure theory.

What is the point of $\pi$-systems and
$\mathcal{D}$ / Dynkin /
$\lambda$-systems?
I am an analyst in the process of consolidating my measure theory knowledge before moving on to ...

**7**

votes

**5**answers

3k views

### The Fundamental Theorem of Calculus in Lebesgue Theory

Dear all,
I am interested to what extent the famous identity
$\int_a^b f'(x) \ dx=f(b)-f(a)$
is true for a function $f:[a,b]-> \mathbb C$ continuous on $[a,b]$ and differentiable on $(a,b)$. One ...

**3**

votes

**2**answers

428 views

### A question about the Kakeya problem

Besicovich proved a long time ago that a straight line segment of fixed length could be rotated 360
degrees within a subset S of the Euclidean plane such that $M(S)$ is arbitrarily small-where M is ...

**5**

votes

**3**answers

888 views

### Finitely additive translation invariant measure on $\mathcal P(\mathbb R)$

We know that a countably additive translation invariant measure with $\mu([0,1]) = 1$ cannot be defined on the power set of $\mathbb R$. This is because $[0,1]$ can be partitioned into countably many ...

**6**

votes

**1**answer

259 views

### Measurable subgroups.

Let $G$ be a compact connected topological group and let $H$ be a subgroup of $G$. Suppose that $H$ is measurable with respect to the normalised Haar measure $\mu$ on $G$. Do we necessarily have ...

**5**

votes

**3**answers

2k views

### Are there sigma-algebras of cardinality $\kappa>2^{\aleph_0}$ with countable cofinality?

A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $\sigma$-algebras. The only proof that I know is via a contradiction argument which ...

**6**

votes

**2**answers

496 views

### Percolation Model and Complex Probabilities

Let $d>0$ be an integer and consider the first neighbors independent bond percolation model in $\mathbb Z^d$, where each edge is open with probability $p\in[0,1]$.
I would like to know, if can we ...

**7**

votes

**2**answers

2k views

### approximate a probability distribution by moment matching

Suppose we want to approximate a real-valued random variable $X$ by a discrete random variable $Z$ with finitely many atoms. Suppose all moments of $X$ is finite. We want to match the moments of $X$ ...

**5**

votes

**4**answers

3k views

### Difference between measures and distributions

On the one hand, Wikipedia suggests that every distribution defines a Radon measure:
http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions
On the other hand, Terry Tao ...

**7**

votes

**1**answer

852 views

### Why is 3 a bad constant in the Vitali covering lemma?

Hi,
recently I had to do with the Hardy-Littlewood maximal function and we used there the Vitali covering lemma with constant 5. Then, given an advice, I proved it with constant k>3. But I cannot ...