48
votes

Accepted

### How decreasing can a bijection $f:\mathbb{N}\to\mathbb{N}$ be?

A fun problem. The answer is $3/4$.
Lower bound: We first make some relaxations to the problem.
Firstly we observe that we can relax the requirement that $f$ be bijective to $f$ being injective, ...

40
votes

### How decreasing can a bijection $f:\mathbb{N}\to\mathbb{N}$ be?

This problem arises in Section 4.3 of my PhD thesis in connection with a Ramsey-type result for infinite tournaments. The answer of $3/4$ is also confirmed by Theorem 4.5 in that section, and the ...

15
votes

### Is there a name for finite unions of intervals?

The discussion in the comments has triggered my curiosity: I've done a little bit of research and I almost immediately stumbled upon the Wikipedia entry on the Peano-Jordan measure. There I found ...

11
votes

Accepted

### Uncountable collections of distinct subsets of an interval (existence)

My comment reposted as an answer:
If the continuum hypothesis holds, then we can give a well order $\prec$ to $\mathbb{R}$ isomorphic to the first uncountable ordinal. And then for each $j\in[-1,1]$ ...

9
votes

Accepted

### Min–max reversing bijections $f:\mathbb{N}\to\mathbb{N}$

$\newcommand\N{\mathbb N}$No, it is not possible to have $\mu_{[\N]^2}\big({\operatorname{rev}(f)}\big) = 1$.
Given the function $f$, we will say that $n\in\mathbb{N}$ is good if $f(n)<f(k)$ for ...

7
votes

### Is there a name for finite unions of intervals?

People working with conformal nets use the term
multi-interval to denote finite disjoint unions of intervals.
https://www.ms.u-tokyo.ac.jp/~yasuyuki/klm3.pdf

6
votes

Accepted

### Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?

Fubini's theorem applied to $|f|$ tells us that the function $g : \Omega \to [0,\infty]$ defined by $g(\omega) = \int_E |f(\omega, x)|\,d\mu(x)$ is measurable with respect to $\mathcal{A}$, and $D$ is ...

6
votes

Accepted

### Sparse "bijection-proof" subsets of $[\mathbb{N}]^2$

Note that $\left\{\{1,a\} \mid a \in \mathbb{N}_{>1}\right\}$ is bijection-proof and $\mu_{\left[\mathbb{N}\right]^2}\left(\left\{\{1,a\} \mid a \in \mathbb{N}_{>1}\right\}\right) = 0$.

6
votes

Accepted

### Shrinking and expanding pairs in bijections $\varphi:\mathbb{N}\to\mathbb{N}$

One has $\max\big\{\mu\big(\exp(\varphi)\big) :\varphi \in S_{\mathbb{N}}\big\}=1$, as mentioned by Emil Jeřábek in the comments.
We can also create a function $\varphi$ with $\mu\big(\text{shr}(\...

5
votes

### Is there a name for finite unions of intervals?

As I see it, it would be preferable a simple term not referring to more advanced characterisations of more general objects (“Finite homology subset”? “Semialgebraic subset”? “Submanifold with ...

5
votes

### Is there a name for finite unions of intervals?

For a non-standard terminology:
In my own analysis lecture notes (sorry, not in a good enough stage to share too widely on the internet at large), I defined a tile in $\mathbb{R}^n$ to be a set of the ...

4
votes

Accepted

### Approximate a non-negative function which is measurable in product $\sigma$-algebra

Take $(E,\mathcal E)=(\Omega,\mathcal G)= \mathbb R$ with the Borel $\sigma$-algebra, and let $C$ be a subset of the diagonal $\Delta\subset \mathbb R^2$. If for a linear combination $g$ of rectangles ...

3
votes

### Sufficient conditions for the space of Radon measure to be a Banach space

If $X$ is a completely regular space, then $M(X)$ is identifiable with the dual of $C^b(X)$ when provided with the so-called strict topology and so is a Banach space. The latter topology was ...

3
votes

### Let $D$ be the set of those $\omega \in \Omega$ such that $f(\omega, \cdot)$ is $\mu$-integrable. Is $D$ measurable?

$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bT}{\mathbb{T}}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bE}{\mathbb{E}}
\newcommand{\bF}{\mathbb{F}}
\newcommand{\bD}{\mathbb{...

3
votes

Accepted

### How to check that the surface measure is the weak limit of $\delta^{-1}\mathcal{L}^n|_{B(0,1+\delta)\setminus B(0,1)}$?

$\newcommand\de\delta\newcommand\si\sigma\newcommand\L{\mathcal L}\newcommand\R{\mathbb R}$Take any continuous function $f\colon\R^n\to\R$. Then $f$ is uniformly continuous on $B(0,2)$. Let $\mu_\de:=\...

3
votes

Accepted

### Is there $\varepsilon \in (0, 1)$ such that $\sup_{t \in [0, \varepsilon]} [\ell_t]_\beta < \infty$?

Pick a sequence of functions $g_n$ where (i) $\|g_n\|_\beta=n$ and (ii) $\|g_n-\mathbf 1\|_\infty\le 3^{-n}$. For example
$$
g_n(x)=\begin{cases}
1+n(a_n-x)^\beta&\text{if $0\le x\le a_n$};\\
1&...

3
votes

### Uncountable collections of distinct subsets of an interval (existence)

Take any sequence $(A_n)$ of measure independent subsets of $[-1, 1]$ of measure $1$, e.g.
$$\begin{align*}
A_1 & = [-1, 0), \\
A_2 & = \left[ -1, -\frac{1}{2} \right) \cup \left[ 0, \frac{1}{...

2
votes

Accepted

### Is projection of a closed subspace Borel?

It is a Borel set. Indeed, let $P: E \times E \to E$ be projection onto the first coordinate. We need to show $P(D_2) \subset E$ is Borel. $P$, restricted to $D_2$, induces a bounded linear map $Q: ...

2
votes

Accepted

### Sufficient condition for uniform convergence of the Stieltjes transform

Weak $*$-convergence $\mu_n\to\mu$ (that is, $\int f\, d\mu_n\to\int f\, d\mu$ for all $f\in C[0,1]$) is equivalent to the locally uniform convergence $G_n(z)\to G(z)$ on $z\in\mathbb C^+$.
This is a ...

2
votes

Accepted

### Approximation on $H^1_0(B)$ and cut-off functions

Do it first for the half-space $\{x_n >0\}=\Sigma$. If $u$ vanishes at the boundary then
$u(x',x_n)^2=2\int_0^{x_n} uD_n u$ and so ($\Sigma_\delta=\{0 <x_n <\delta\}$)
$$
\int_{\Sigma_\delta} ...

1
vote

Accepted

### Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC?

Here is a proof that only uses countable choice. It is taken from Fremlin, Measure Theory, Volume 5, Number 566F. The chapter 56 of this multi-volume work is a great source on the discussion of choice ...

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