Monotone Classes

Definition

The limit of an increasing sequence of sets is the family union of the sequence. The limit of a decreasing sequence of sets is the family intersection of the sequence.

A monotone limit of an sequence of sets is the limit of a monotone sequence.

A monotone class is a subset system in which monotone limits of monotone sequences of distinguished sets are distinguished. We call the distinguished sets a monotone class.

Notation

Let $A$ a non-empty set with partial order $\preceq$. Let $(A, \mathcal{A} )$ be a subset space on $A$.

Let $\seq{A}$ be an increasing or decreasing sequence in $\mathcal{A}$. We denote the limit of $\seq{A}$ by $\lim_{n} \seqt{A}$.

If $\seq{A}$ is increasing, $\lim_n \seqt{A} = \cup_{n} \seqt{A}$. If $\seq{A}$ is decreasing, $\lim_n \seqt{A} = \cap _{n} \seqt{A}$.

If $(A, \mathcal{A} )$ is a monotone space, then for all monotone $\seq{A}$ in $\mathcal{A} $, $\lim_{n} \seqt{A} \in \mathcal{A} $. In this case, $\mathcal{A} $ is a montone class.