Monotone class. A class F of subsets of a space is a field if it contains the whole space and is closed under complementation and under finite unions; a classMis monotone if the union and intersection of every increasing and decreasing sequence of sets of M is again in M. The smallest monotone class M0 containing a given field F coincides with the smallest σ-field A containing F. [One proves first that M0 is a field. To show, for example, that A ∩ B ∈ M0 when A and B are in M0, consider, for a fixed set A ∈ F, the class MA of all B in M0 for which A ∩ B ∈ M0. Then MA is a monotone class containing F, and hence MA = M0.

Thus A ∩ B ∈ MA for all B. The argument can now be repeated with a fixed set B ∈ M0 and the class MB of sets A in M0 for which A ∩ B ∈ M0. Since M0 is a field and monotone, it is a σ-field containing F and hence contains A. But any

σ-field is a monotone class so that also M0 is contained in A.]