> It doesn't help that the whole section on sigma algebras is somewhat irrelevant, since he doesn't really explore measure theory as the basis for modern probability.

Christ, practically all of measure theory is irrelevant for applied work, in much the same way that an engineer shouldn't care about the definition of a real number.

There's a model of real analysis due to Solovay that used the axiom of dependent choice instead of the full axiom of choice. In the Solovay model, all sets are measurable. Thus any results that require measure theory inherently depend on the axiom of choice.

I'd be worried if I was relying on Choice as an applied scientist.

Edit: same goes for Lebesgue vs. Riemann integration. To quote Richard Hamming: Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.