But don't want to do it there because I don't feel like taking the time to explain the details, which I would have to if queried by Ollie.
Basically, in practice we do not need any measure theory.
You can treat a random variable as the output of a non-deterministic machine. Every time you query that machine, you get some output that may differ from the previous one. Also, you can construct a machine that is essentially a program that can use previously given or constructed machines. So for example you can construct a machine that rolls a die until it shows 6 (the dice is itself a machine), and then outputs the number of rolls taken.
In such a framework, one can still analyze almost all practical probabilistic processes.
Without any measure theory; just reasoning about conditional probabilities according to the program structure.
Does what I say make sense? It's not the conventional way of doing probability theory, but works well in many situations.
@user21820 I wasn't suggesting that the only way to tackle the problem was via measure theory, only that if they knew some measure theory, there was a ready explanation.
@XanderHenderson Oh okay. I couldn't tell because there weren't any follow up comments.
And technically, even the measure theory explanation is not really ready, because it basically just says that you have to set up a probability space before "random set of reals" even makes sense.
@user21820 There weren't any follow up comments because the one answer that I feel moderately comfortable giving is from a measure theoretic point of view, and the asker would not have been able to use that (I think). It wouldn't have been a very good answer, anyway.
@XanderHenderson Anyway I think measure theory per se doesn't really help one to construct random objects, in the sense that merely defining them can be a difficult thing in itself. For example, 2 years ago I couldn't figure out how to construct the following:
I saw this question, and was wondering what is the best way to describe a random distribution of points in the plane such that the expected number of points in any region of unit area is $c$, where $c$ is a given positive constant, and such that the number of points in every pair of disjoint regi...
Thanks to Michael and Shalop for the core ideas, here is a rigorous definition and proof.
$
\def\nn{\mathbb{N}}
\def\zz{\mathbb{Z}}
\def\rr{\mathbb{R}}
\def\pp{\mathbb{P}}
\def\ee{\mathbb{E}}
\def\ii{\mathbf{1}}
\def\wi{\subseteq}
\def\none{\varnothing}
\def\t{\text}
\def\lsum{{\large\sum}}
\def\...
And note that the points where I used MCT do not really need MCT if the 'query' sets one is interested in are simpler (say Jordan-measurable).
This is the Realm of Simply Beautiful…
This is the Realm of Simply Beautiful…