TITLE: Sigma algebra generated by a set
QUESTION [6 upvotes]: When determining the smallest sigma-algebra generated by a finite collection of sets (and hence the smallest algebra containing that collection), is there any faster way to do this than by direct computation? On a related note, is there a criterion for selecting a set which generates the finest sigma-algebra (that is, the power set)?

REPLY [2 votes]: 1) take intersections and complementations until you get the atoms. Then these atoms will generate the sigma algebra. Moreover, the sigma algebra is the power set of the atoms.
2) if in the process of taking intersections, you get to sets which form a partition of the "sample set" consisting of atoms, then you are done. So take intersections until you obtain such a partition.  That's fun. Such a partition determines uniquely the subalgebra.
3) it results form 1) that the sigma algebra has cardinal $2^n$, where n is the number of atoms; so the cardinal should be of the form 2,4,8,16 and so on. That's a nice why to check if you got it right.