I actually wrote this document earlier this year, but was not able to make this blog post as I did not have any internet connection at home (special thanks to the Telekom, who has taken since November to install a simple fiber optic connection in my apartment, and was unable to do so after 5 attempts!)
In any case, I provide here an explanation of the monotone class theorem. It was written in an attempt to illustrate the differences between (\sigma)-algebras and monotone classes to me, as well as prove the theorem.
It seems that I was unable to complete the section on applications, but that is hopefully alright. I will finish it if I require it to deepen my understanding.