  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.
