My beef with this is precisely that it doesn’t use the general background of measure theory (sigma-algebras, measurable functions, etc.) and you’re going to need that background to do useful things.
And Bartle covers them, but later. Section 19.
The gauge integral approach doesn’t give you the tools to generalize to scenarios like Brownian motion where you need to construct different measures;
I have on my desk right now Steven Shreve’s Stochastic Calculus for Finance II, and the construction of the Wiener process is in slightly different language the limit of a sequences of functions defined on a sequence of tagged partitions. I’m just now learning stochastic-flavored things, so I don’t know if this is canonical.
also, the gauge integral doesn’t come with a lot of nice convergence theorems the way the Lebesgue measure does.
Section 8 covers the main three (Monotone, Fatou’s Lemma, and Dominated Convergence).
Also, there’s some virtue to just being familiar with the definitions and concepts that everybody else is.
Sunken cost fallacy.
(It’s not just mathematicians “refusing to update.” I know for sure that economists, and potentially people in other fields, speak the language of standard measure theory. But maybe it’s not everyone. What are you using measure theory for?)
I’m a grad student doing PDEs. I think there are two issues that need to be separated here. The first is pedagogical. People doing probability theory do need to learn measure theory eventually, yes. Royden takes this same approach—show the Lebesgue measure on R to start and then progress to abstract measure spaces. Unfortunately he fills the middle bits between chapters five and nine (I think) with a lot of topology.
The second is practical. There are more gauge-integrable functions than Lebesgue-integrable functions. There are nice lemmas for estimating gauge integrals, and they tend to be slightly more concrete.
I also rank Halmos higher than Cohn in terms of measure theory books. Your mileage may vary.
I also rank Halmos higher than Cohn in terms of measure theory books
Halmos, by the way, is a top-notch mathematical author in general. Every one of his books is excellent. Finite-Dimensional Vector Spaces in particular is a classic.
I think that’s the only book I kept from my Maths degree. In hardback, too. I have lent it to a colleague and keep a careful eye on where it is every couple of weeks...
And Bartle covers them, but later. Section 19.
I have on my desk right now Steven Shreve’s Stochastic Calculus for Finance II, and the construction of the Wiener process is in slightly different language the limit of a sequences of functions defined on a sequence of tagged partitions. I’m just now learning stochastic-flavored things, so I don’t know if this is canonical.
Section 8 covers the main three (Monotone, Fatou’s Lemma, and Dominated Convergence).
Sunken cost fallacy.
I’m a grad student doing PDEs. I think there are two issues that need to be separated here. The first is pedagogical. People doing probability theory do need to learn measure theory eventually, yes. Royden takes this same approach—show the Lebesgue measure on R to start and then progress to abstract measure spaces. Unfortunately he fills the middle bits between chapters five and nine (I think) with a lot of topology.
The second is practical. There are more gauge-integrable functions than Lebesgue-integrable functions. There are nice lemmas for estimating gauge integrals, and they tend to be slightly more concrete.
I also rank Halmos higher than Cohn in terms of measure theory books. Your mileage may vary.
Halmos, by the way, is a top-notch mathematical author in general. Every one of his books is excellent. Finite-Dimensional Vector Spaces in particular is a classic.
I think that’s the only book I kept from my Maths degree. In hardback, too. I have lent it to a colleague and keep a careful eye on where it is every couple of weeks...