I'm auditing a course called "Topics in Analysis", and in class we mentioned a theorem that didn't make sense to me. I don't fully remember how it goes, but it was something like:
Theorem (half-remembered, probably not the correct formulation): The limit of an $L^1$-bounded Martingale can be uniquely decomposed into the sum of three pieces:
- an $L^1$ function
- a countable collection of Dirac masses
- a continuous Borel measure
The reason this confused me was, first of all, how are we adding functions and measures together? Are we implicitly converting functions to measures so that they have the same type? And if we are, why is category 1 disjoint from category 3? (I say they must be disjoint because otherwise we break uniqueness by subtracting f from the first piece of the decomposition and adding back f, or at least its measure version, into the third piece.)
So I'd greatly appreciate it if someone could either explain this to me, or provide me with a reference so I could read about this, because I've clearly misunderstood something here.
(I tried asking the teacher about it after class, but he kept on just stating that 1 and 3 are disjoint and repeating the statement of the theorem, without answering why or how or addressing my question. It was very frustrating)
EDIT: Oh, I should also say that I tried Googling this and couldn't find anything.