The monotone class theorem states that for any algebra of sets $\cal A$ one can construct the smallest monotone class generated by this class ${\cal M}(\cal A)$. This smallest monotone class is also the smallest sigma-algebra $\Sigma(\cal A)$ generated by $\cal A$, and ${\cal M}(\cal A)=\Sigma(\cal A)$.
However, I noticed that in applications of the theorem in various mathematical proofs I have studied authors ignore the fact that the monotone class should be the smallest.
For example, assume that the goal is to prove that a certain class of sets, e.g. $\cal B$, is a sigma-algebra. Authors just prove that this group of sets is a monotone class, and then by invoking the monotone class theorem, without showing that $\cal B$ is indeed the smallest monotone class, just conclude that this class has to be also a sigma-algebra.
Is it in general possible to ignore the "smallest" requirement when using the monotone class theorem, or are there circumstances one has to be aware of that make such use possible?
EDIT: The background to the question can be found here. The post discusses a theorem where the "smallest" requirement is being ignored. It provides an example.
