I am reviewing the way in which Ash et al. (1999) introduces the extension of measures.
My Question
Is it an accurate characterization to say the defining difference between a generic field and a $\sigma$-field is that the limiting sets of an increasing or decreasing sequence belongs to the field? This is a long way to say a field is closed under finite union while a $\sigma$-field is closed under countable union. So is my characterization same? I am thinking in the same vein of why finite additivity plus continuity implies countable additivity, and continuity requires the limiting sets to belong to the field.
Reference:
$\textit{Probability and Measure Theory}$ (Robert B. Ash and Catherine A. Doleans-Dade), Harcourt/Academic Press, 1999.
Best Answer
Yes, a field closed under increasing limits is always a $\sigma$-field, and this is equivalent to saying that a $\sigma$-field is a field that is closed under countable unions. To see this take any sequence ($A_n$) of members of the field. Then construct an increasing sequence by setting $B_n$ equal to the union of the first $n$ members of the sequence ($A_n$). The sets $B_n$ are also members of the field since they are finite unions. Then the limit of $B_n$ is the countable union of all $A_n$ and belongs to the ($\sigma$-)field by assumption.
And of course a $\sigma$-field is closed under increasing limits, since by definition the limit is a countable union of elements of the field.