> I think there is a point of diminishing returns.

Yes, there is a big question about what to learn, about how much to invest in such things.

> Baby Rudin I'm dubious about. But Royden and big Rudin (both of which you recommended) I have certainty about.

But Baby Rudin is a prerequisite to Royden and big Rudin.

I'm sorry, but probability, stochastic processes, and mathematical statistics were junk for me until I went at them via measure theory.

I floundered terribly with random variables until I saw the measure theory definition; it's terrific: Go take 10,000 measurements. Now have the values of 10,000 random variables. Any 10,000 measurements at all. So far, no concept of 'randomness' at all. So, random variables are very general things and, e.g., handle even deterministic processes as a special case.

E.g., sufficient statistics is just an application of the Radon-Nikodym theorem, and a total train wreck to do otherwise. Yes, order statistics are always sufficient, maybe nice to know in 'data mining'. That sample mean and sample variance are sufficient in the Gaussian case is mind blowing; nice opportunity for 'data compression'!

E.g., measure theory and the Radon-Nikodym theorem define conditional expectation, that is, under mild assumptions, E[Y|X] = f(X) for some measurable f. Then easily f(X) is the best non-linear least squares approximation of Y. Nice.

Further, if 'cross tabulate' Y on X, then have a discrete approximation to E[Y|X] which shows that cross tabulation is a discrete version of the best non-linear approximation of Y given X.

Measure theory permits working with all the forms of convergence of random variables, especially strong convergence, at least awkward to do otherwise.

Then martingale theory makes little sense without measure theory.

Measure theory, the Kolmogorov extension, shows that we really can have a collection of random variables with desired properties.

Measure theory is crucial in even defining E[Y|U(t), t <- a] since we are conditioning on an uncountably infinite collection of random variables. Measure theory, then, shows that we can replace U(t), t <= a by the sigma algebra they generate and, then, condition on the sigma algebra. Cute.

Constructions such as

E[Y|U(t), t <= a]

are crucial in the nice qualitative, axiomatic definition of the Poisson process.

Similarly for independence of two collections of random variables where each collection has uncountably infinitely many random variables.

Measure theory was crucial in the standard results of ergodic theory.

I wrote a paper on anomaly detection in server farms and networks, and the key idea in the paper was a finite group of measure preserving transformations lifted roughly from ergodic theory.

Via measure theory we can show that the space of real valued L^2 random variables is complete, and, thus, a Hilbert space, which continues to blow my mind that any such thing could be true. I'd also like to have locally compact, but that's a bit much to hope for!

The Doob decomposition shows that every stochastic process is the sum of a martingale and a predictable process, all measure theory!

It's tough enough to believe in probability with the measure theory foundations; otherwise, I couldn't swallow the stuff!

There is a broad point: Maybe OP wants to know what to learn for the applications of the future. Then what current CS profs know is not necessarily very relevant!