On the existence of a minimal generating set for $\sigma$-algebras

TL;DR

This paper investigates whether every $\sigma$-algebra has a minimal generating set, providing an answer for specific cases like partition-generated and standard measurable spaces, while leaving the general case unresolved.

Contribution

It formulates the problem of minimal generating sets for $\sigma$-algebras and solves it for particular classes of spaces, advancing understanding in measure theory.

Findings

Existence of minimal generating sets proven for partition-generated spaces

Existence of minimal generating sets proven for standard measurable spaces

The general case remains an open problem

Abstract

Does there exist for any $\sigma$-algebra a minimal (with respect to inclusion) generating set? We formulate this problem and answer it in the very special instance of partition generated and standard measurable spaces, the general case remaining open.