Difference between pure probability and measure-theoretic probability What is the difference between probability theory when it is and isn't based on measure theory?
My university teaches it with measure theory, and I am enjoying it so far, but am curious as to what I am missing out on. Is "pure" probability theory worth taking a look at, or will it be a waste of time due to already having things covered with measure-theoretic PT?
 A: Probability theory based on measure theory is not less pure than purely mathematical probability not requiring understanding of measure theory.
If one regards Kolmogorov as God's last prophet in the field of probability, as some mathematicians in effect do, then measure-theoretic probability is probability.  Probability courses not involving measure theory are intended for people who don't know measure theory.  You can say quite a lot about probability that can be understood without knowledge of measure theory and it's perfectly possible that a lot of good stuff is in that other course that's not in the measure-theoretic course.
Some people reject the idea that countable additivity should be taken as axiomatic.  Then they can talk about a uniform distribution of sorts on the set of all integers.  Inequalities for Stochastic Processes by Lester Dubins and Leonard Jimmy Savage dispenses with countable additivity for technical reasons.
One can also argue that the following question belongs to probability theory but not to mathematics (and that would mean probability theory is a science relying heavily on mathematics, just as physics does, but is not actually a field within mathematics, just as physics is not): Why should the conventional axioms of probability apply when probabilities are taken to be epistemic, i.e. assigned to uncertain propositions and expressing logically justified degrees of belief, rather than relative frequencies?  A number of answers to that have been published, including Bruno de Finetti's thought-experiments on gamling strategy and Richard T. Cox's book Algebra of Probable Inference. (And my own paper titled "Scaled Boolean Algebras" in the 2002 volume of Advances in Applied Mathematics (this was before the Elsevier boycott).)
Occasionally a mathematician rejects outright (as William Feller did) the idea that such epistemic probability theory has any validity or utility.  An important thing to notice about that is that such a rejection is in no sense a mathematical proposition, susceptible of proof or disproof.
The proposition that probabilities should be taken to be epistemic, regardless of whether that means taking them to be subjective (on the one hand) or (on the other hand) logical is called Bayesianism.  Bayesian methods of statistical inference are invulnerable to some pathologies that can afflict what are called frequentist methods, but you encounter the hard problem of assignment of prior probabilties, and that is not a mathematical problem.  Lest anyone think that that means subjectivity is inherent in Bayesian methods but not in frequentist methods, let us note that the choice of maximum probabilities of Type I and Type II errors that one is willing to tolerate in frequentist hypothesis testing is itself a subjective economic decision (unless you take the philosophical view, as some do, that economic decisions are not basically subjective).
A: Measure-theoretic probability is the usual sort of probability.  The alternative would be to take a "lite" version where you have to talk about continuous and discrete stuff separately.
