Measure finite topology on the ring of measurable functions

TL;DR

This work defines the measure-finite topology $F_\mu$ on the ring $\mathcal{M}(X,\mathcal{A},\mu)$ of real-valued measurable functions and analyzes how the underlying measure $\mu$ governs fundamental topological properties. By showing that the $F_\mu$-topology sits between the measure-theoretic $U_\mu$ and $m_\mu$ topologies, the authors connect connectedness, first/second countability, and various cardinal invariants to measure-theoretic notions such as atomicity and hemifiniteness. A central result is that the path, connected, and quasi components of $\underline{0}$ coincide and equal $\bigcap_{G>0} L_G^\infty$, with precise equivalences: MF is path connected (and a topological ring/vector space) precisely when $\mu$ is atomic of a special type; MF is first countable iff $\mu$ is hemifinite; and MF is second countable precisely when $\mu$ is hemifinite and the topology satisfies the countable chain condition. The paper further develops the controlling number of the sigma-algebra, equating key cardinal invariants (tightness, $\pi$-character, and character) to $cn(\mathcal{A})$, and establishes a suite of equivalent conditions for second countability, Lindelöfness, separability, and CCC under hemifiniteness, thereby linking measure-theoretic structure to topological behavior of function rings.

Abstract

Let $\mathcal{M}(X,\mathcal{A},μ)$ be the ring of all real-valued measurable functions constructed over a measure space $(X,\mathcal{A},μ)$. A topology on $\mathcal{M}(X,\mathcal{A},μ)$, called the {$F_μ$-topology} weaker than the { $U_μ$-topology} is introduced. It is realized that the {component}, the {quasi component} and the {path component }in this {$F_μ$-topology} are identical. It turns out that the {$F_μ$-topology} on $\mathcal{M}(X,\mathcal{A},μ)$ becomes {connected} if and only if it is {path connected} if and only if $μ$ is an {atomic measure} of a special type. It is also proved that the {$F_μ$-topology} is {first countable} when and only when $μ$ is a {hemifinite measure.} Finally, it is shown that the {second countability} of the {$F_μ$-topology} is equivalent to the {hemifiniteness} of the measure $μ$ together with the {countable chain condition} of the {$F_μ$-topology}.

Theorems & Definitions (48)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5
• Lemma 3.1
• proof
• Theorem 3.2