I have always struggled in understanding probability theory, but since coming across the measure theoretic approach it seems so much simpler to grasp. I want to verify I have a couple basic things. So say we have a set χ. Together with a σ-algebra κ on χ, we can call (χ,κ) a measurable space. We can define a measure on χ, call it M, that follows the basic requirements (non-negativity, countably additive, the measure of the empty set is 0, etc). Now, (χ,κ,M) is a measure space. Now, getting to probability theory, if M(χ) < ∞, M is a finite measure and in particular, if M(χ)=1, we can consider M as a probability measure. If we're talking about a probability measure, we can consider χ as our sample space, and forall E in κ, E is an event. Furthermore, we can the measure space (χ,κ,M) a probability space. If M is a probability measure, what it does is it assigns to each event E in κ, a probability, which is the chance it will happen. (that's why M(χ)=1, this is like saying, if I get some feasible event E as an outcome, what's the probability that E is any of the feasible events in χ, well obviously, this is a 100% chance, while M(∅)=0, what's the probability that E isn't any of the possible events in χ? Well, no chance of that happening since we've established before hand that E is feasible). So here is my confusion: understanding the random variable, R. From what I understand, the analogue to measure theory is the random variable is a measurable function. For example, like the counting measure, or the Lebesgue measure. A measurable function is a function R: S1 --> S2 where (S1,Ω1,M1) and (S2,Ω2,M2) are measure spaces, so that if E in Ω2 is an event in Ω2 (the σ-algebra on S2), then R-1(E) in Ω1 (its pre-image under R is an event in the σ-algebra on S1). I have difficulty understanding intuitively what this is in a probability sense. What are the two measure spaces we would be mapping between? I feel like with probability, I"m only looking at one measure space: you know, for example, the population of a country, this could be my sample space, where as, the events could correspond to different possible ages of the residents, so an event 1 might be (person A is 25, person B is 30, etc. etc.). What is this "second measurable space" I would be dealing with when establishing a random variable?