I'm an engineering student who has taken one undergraduate course in probability theory, but that's all my exposure so far. I'm trying to get into machine learning and need to develop more of a background for this purpose.
I hear about different types of probability theory and where it builds from, but the connections between these are extremely hazy to me. For example, I've heard of probability theory being grounded on top of measure theory, but have also heard of it based on logic, have heard the terms Kolmogorov and de Finetti probability theory, but can't make sense of what's going on here. Is there different factions of probability theorists who disagree with each other or do they all play a part in some coherent framework? Would anyone give me a beginner's synopsis of the relationships and layout between these things?
Kolmogorov probability theory defines the probability measure on $\sigma$-algebras, and the measure is countable additive. This is what pretty much everybody uses.
De Finetti defines the the probability measure on algebras, and the measure is only finitely additive. It is an alternative theory, but beyond you would actually need.
My suggestion would be to start with basic books, such as Sheldon and Ross, and upgrade to more advanced things (Grimmet is good for engineering students, Billingsley is good too) only after you have become confortable with conditioning, independence, joint pdf's,etc.