On the Construction and the Cardinality of Finite $σ$-Fields

In this note, we first discuss some properties of generated $σ$-fields and a simple approach to the construction of finite $σ$-fields. It is shown that the $σ$-field generated by a finite class of $σ$-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 $σ$-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 $σ$-fields. Finally, an application of our results to statistics, with regard to independence of events, is pointed out.

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