0
votes
0answers
49 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
0answers
79 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
0answers
56 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
0answers
163 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
2answers
180 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
1answer
184 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
1answer
184 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
2answers
349 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
2answers
247 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
1answer
136 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
1answer
265 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
1answer
255 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
1answer
650 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
1answer
751 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
0answers
254 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
0answers
690 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 ...
0
votes
1answer
423 views

Extending Continuous Sublinear maps on dense subsets of a Banach space

Suppose X' and Y are Banach spaces and X is a linear subspace dense in X'. Let T be a continuous map of X to Y satisfying: (1) ||T(x+y)|| is less than or equal to ||T(x)||+||T(y)||. Please prove ...
9
votes
3answers
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
2answers
288 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
1answer
362 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
4answers
797 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 ...
9
votes
0answers
455 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 ...
43
votes
23answers
14k views

Why should one still teach Riemann integration?

In the introduction to chapter VIII of Dieudonné'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
1answer
657 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
2answers
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
1answer
628 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
3answers
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
2answers
547 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
11answers
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
4answers
816 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
2answers
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
0answers
219 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 ...
12
votes
4answers
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
2answers
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
5answers
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
2answers
419 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
3answers
866 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
1answer
257 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
3answers
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
2answers
492 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 ...
6
votes
2answers
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
4answers
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
1answer
822 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 ...