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.(adsbygoogle = window.adsbygoogle || []).push({});

So say we have a set χ. Together with a σ-algebra κ on χ, we can call (χ,κ) ameasurable space. We can define ameasureon 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 ameasure 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 aprobability measure. If we're talking about a probability measure, we can consider χ as our sample space, and forall E in κ, E is anevent. Furthermore, we can the measure space (χ,κ,M) aprobability 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 isany of the feasible eventsin 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 ameasurable function. For example, like the counting measure, or the Lebesgue measure. A measurable function is a function R: S_{1}--> S_{2}where (S_{1},Ω_{1},M_{1}) and (S_{2},Ω_{2},M_{2}) are measure spaces, so that if E in Ω_{2}is an event in Ω_{2}(the σ-algebra on S_{2}), then R^{-1}(E) in Ω_{1}(its pre-image under R is an event in the σ-algebra on S_{1}). 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?

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# Basic random variable question - measure theory approach

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