I have been reading through Wikipedia pages, and I'm still really confused. What is the difference between "algebra of random variables" and "probability space."? Are they just different words for describing the same thing, or are there fundamental differences?
At the bottom of the Probability Space page, it says that a prodability measure is a probability density of a random variable. However, near the top, it says, "The prominent Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. Nowadays alternative approaches for axiomatization of probability theory exist; see “Algebra of random variables”, for example." Which suggests two me that they are two competing approaches/theories.
Can anyone explain to me how these terms and ideas fit together -- What are the primary conceptual differences between the two and the advantages/disadvantages of each?
I know that this question probably won't make sense to someone who actually understands what the terms really mean -- so please try to imagine a beginner who is just trying to make sense of the field, and understand why there are different terms that seem to apply to the same concepts.