# Tag Info

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, ...
• 112k

### 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 ...

### 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 ...
• 5,832
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 $j\in[-1,1]$ ...
• 10.2k
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 ...
• 10.2k

### 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
• 42.7k
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 ...
• 29.5k
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 $\mu_{\left[\mathbb{N}\right]^2}\left(\left\{\{1,a\} \mid a \in \mathbb{N}_{>1}\right\}\right) = 0$.
• 911
Accepted

• 1,063
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• 1,387
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 ...
• 23.6k
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} ...
• 5,365
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 ...
• 273

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