On the Construction and the Cardinality of Finite $\sigma$-Fields

TL;DR

This paper explores properties, construction methods, and cardinality calculations of finite sigma-fields, providing explicit formulas, examples, and an algorithm, with applications to statistical independence.

Contribution

It introduces a simple approach to constructing finite sigma-fields and derives explicit cardinality ranges, including an algorithm for specific cases.

Findings

The sigma-field generated by finite atoms equals that from the induced partition.

Explicit range of cardinalities for such sigma-fields is obtained.

An algorithm for calculating the exact cardinality of certain finite sigma-fields is presented.

Abstract

In this note, we first discuss some properties of generated $\sigma$-fields and a simple approach to the construction of finite $\sigma$-fields. It is shown that the $\sigma$-field generated by a finite class of $\sigma$-distinct sets which are also atoms, is the same as the one generated by the partition induced by them. The range of the cardinality of such a generated $\sigma$-field is explicitly obtained. Some typical examples and their complete forms are discussed. We discuss also a simple algorithm to find the exact cardinality of some particular finite $\sigma$-fields. Finally, an application of our results to statistics, with regard to independence of events, is pointed out.