# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $X$ ...