I'm reading link: Foundations of theory of probability – Kolmogorov and have some questions regarding this historical text. I'm a bit stuck, so any guidance is appreciated. Thanks!
In the axioms below it is stated that "F a set of subsets of $E$" and per Axiom III. "to each set $A$ in F". Therefore it's my understanding field F includes a subset of the power set $\mathcal{P}(E)$. Is this the correct understanding?
He then states in (8) below that the system of sets $\mathcal{U}$ forms a decomposition and the sets $A_1, A_2, …, A_i$ do not intersect. It's my understanding a decomposition is also known as a partition. I'm confused because there are earlier statements that sets $A_i$ form a subset of the power set, in which these sets would possibly intersect, while later in item 8 he states that "the sets, do not intersect, in pairs." Why the conflict or am I missing something?
Finally based on above I'm trying to understand what's expressed on on Page 9 below. What is independence? Can anyone guide through this page? I understand that each experiment has a set of decompositions, however am confused because based on earlier statements I assume the decompositions to be the same every time, specifically for the power set.
Can anyone explain the following form? The confusing part to me, is if the $P(a_1, a_2, …, a_n)$ sets do not intersect, i.e. they are mutually exclusive, there is still probability to the right side of the question. This seems like an inequality.
Best Answer
Yes, here $\cal F$ is a subset of the power set of $E$. In other words, the types of things in $\cal F$ are sets, and the elements of those sets are in elements from $E$. Alternatively, everything in $\cal F$ is in $\mathcal P(E)$ but not necessarily the other way around.
A decomposition $\cal U$ is a special subset of the power set in which the sets in $\cal U$ do not intersect each other at all, and they cover the entire set $E$. Not every subset of the power set is a decomposition, but every decomposition is a subset of the power set.
Finally, independence DOES NOT mean disjoint/mutually exclusive. The definition of independence is exactly that the probability of the intersection equals the products of the probabilities; i.e. if that equation is true, then we can call those experiments "mutually independent", and if the equation is not true, those experiments are not "mutually independent". There is not really an intuitive picture you can draw in your head of when two things are independent.
Lastly, I'm not sure that learning probability theory from Kolmogorov's book is the best idea. I think it would be much easier to learn it from a more modern book with more modern/more intuitive notation first, then come back to Kolmogorov. However, I do not claim to know your situation, so of course this is just my two cents.
Personally I started my probability journey from measure.axler.net/MIRA.pdf (chapter 2,3 and chapter 12 are the key parts). A classic book on pure probability theory would be colorado.edu/amath/sites/default/files/attached-files/…, which I think is pretty good. I don't know what level you want to study the subject at, but these are where I started.