Isometric embedding of ultrametric (non-Archimedean) spaces in Hilbert space and Lebesgue space (Q2751735)

A metric space \((X,d)\) is called ultrametric if its metric satisfies \(d(x,z)\leq \max\{d(x,y), d(y,z)\}\) for all \(x,y,z\in X\). The goal of the paper is to describe and compare two properties of utrametric spaces -- embedding in Hilbert space and Lebesgue space. It is proved that any separable (countable, finite) ultrametric space can be imbedded isometrically in the space \(L(\mathbb{R})\) of Lebesgue measurable subsets of the real line \(\mathbb{R}\) (of Borel measurable subsets, respectively). It is shown that any ultrametric space \(X\) of weight \(\tau\) can be imbedded isometrically in the generalized Hilbert space \(H^\tau\) of the same weight. In particular, for \(\tau= \aleph_0\), \(X\) is imbeddable in classical (separable) Hilbert space \(H\). Comparison is made of these two types of imbeddings as well as metric properties of \(H\) and \(L(\mathbb{R})\). Examples of finite metric spaces which are not embeddable in \(L(\mathbb{R})\) are given. It is also shown that any separable ultrametric space can be imbedded isometrically in the space \({\mathcal L}_p(\mathbb{R})\) of Lebesgue integrable functions on \(\mathbb{R}\) for any \(p>0\).NEWLINENEWLINEFor the entire collection see [Zbl 0969.00058].