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> 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 proba
by soVeryTired 9y ago
> 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.
- psoy 9y agoIt matters very much in computational / quant finance
- soVeryTired 9y agoIt matters by convention, because the textbooks are written that way. My point is that you don't need that level of formal rigour to do applied work. You can derive the Feynman-kac formula via a scaling limit of discrete-time Markov chains. Add some levy process (a.k.a compound Poisson processes) and you're basically done. If you want to be ultra-rigourous in your definitions, then you need measure theory, yes. But even Einstein didn't need that for his description of Brownian motion. If a scaling limit is good enough for him, it's good enough for me.