Let $M(X,𝒜)$ ($M^{*}(X,𝒜)$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X,𝒜)$, let $M_K(X,𝒜)$ be the family of all $f\in M(X,𝒜)$ such that $\mathrm{coz}\left(f\right)$ is compact, and let $M_{\infty }(X,𝒜)$ be all $f\in M(X,𝒜)$ that $\{x\in X:|f\left(x\right)|\ge \frac{1}{n}\}$ is compact for any $n\in \mathbb{N}$. We introduce realcompact subrings of $M(X,𝒜)$, we show that $M^{*}(X,𝒜)$ is a realcompact subring of $M(X,𝒜)$, and also $M(X,𝒜)$ is a realcompact if and only if $(X,𝒜)$ is a compact measurable space. For every nonzero real Riesz map $\varphi :M(X,𝒜)\to ℝ$, we prove that there is an element $x_0\in X$ such that $\varphi \left(f\right)=f\left(x_0\right)$ for every $f\in M(X,𝒜)$ if $(X,𝒜)$ is a compact measurable space. We confirm...

In the theory of nonarchimedean normed spaces over valued fields other than R or C, the property of spherical completeness is of utmost importance in several contexts, and it appears to play the role conventional completeness does in some topics of classical functional analysis. In this note we give various characterizations of spherical completeness for general ultrametric spaces, related to but different from the notions of pseudo-convergent sequence and pseudo-limit introduced by Ostrowski in...

Let K be an algebraically closed complete ultrametric field. M. Krasner and P. Robba defined theories of analytic functions in K, but when K is not spherically complete both theories have the disadvantage of containing functions that may not be expanded in Taylor series in some disks. On other hand, affinoid theories are only defined in a small class of sets (union of affinoid sets) [2], [13] and [17]. Here, we suppose the field K topologically separable (example Cp). Then, we give a new definition...

Let (K,v) be a henselian valued field of arbitrary rank which is not separably closed. Let k be a subfield of K of finite codimension and $v_k$ be the valuation obtained by restricting v to k. We give some necessary and sufficient conditions for $(k,v_k)$ to be henselian. In particular, it is shown that if k is dense in its henselization, then $(k,v_k)$ is henselian. We deduce some well known results proved in this direction through other considerations.

In this article, it will be shown that every $\ell$-subgroup of a Specker $\ell$-group has singular elements and that the class of $\ell$-groups that are $\ell$-subgroups of Specker $\ell$-group form a torsion class. Methods of adjoining units and bases to Specker $\ell$-groups are then studied with respect to the generalized Boolean algebra of singular elements, as is the strongly projectable hull of a Specker $\ell$-group.