On (real) analysis: Bartle’s A Modern Theory of Integration.
Even Bayesian statistics (presumably the killer app for analysis in this crowd) is going to stumble over measure theory at some point. So this recommendation is made with that in mind.
The traditional textbooks for modern integration in this context are (the first chapters of) Rudin’s Real and Complex Analysis and (the first chapters of) Royden’s Real Analysis.
I can’t recommend Rudin because in the second chapter he goes on this ridiculously long tangent on Urysohn’s lemma that makes absolutely no sense to anyone who hasn’t seen topology before. Further, the exercises tend to have a difficulty curve that starts a bit too high for the non-mathically inclined.
Royden is slightly better in this respect. The first four chapters are excellent, but still probably too theoretical. Further, eventually one will encounter measure spaces that aren’t based on the real numbers and the Lebesgue measure, and because of the way Royden is set up the sections on Lebesgue theory and abstract measure theory are separated by a refresher on metric spaces and topology. Unlike the tangent in Rudin, this digression isn’t as avoidable.
My recommendation then corrects for these errors. Bartle’s book works with the gauge integral, which is perfectly compatible with the Lebesgue integral (i.e., they give the same results when both work) but has a more concrete formulation (not requiring any measure theory). I expected that a book taking this route would avoid measures altogether, but this is incorrect—even with the gauge integral questions of measurability come into play, and Bartle’s book covers these adequately.
As an aside, the gauge integral is one example of mathematicians failing to update, in a sense. It’s pretty superior to the Lebesgue integral in terms of conceptual simplicity and applicability, but practically no one uses it.
Hmm. Upvoted for contributing to a good topic but I’m not sure I agree.
I just looked up the gauge integral because I wasn’t familiar with it. For those curious about the debate, here’s the introduction to the gauge integral I found, which has a lot of relevant information. 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. The gauge integral approach doesn’t give you the tools to generalize to scenarios like Brownian motion where you need to construct different measures; also, the gauge integral doesn’t come with a lot of nice convergence theorems the way the Lebesgue measure does.
I don’t find the standard treatment of measure theory especially hard; it takes about a month to understand everything up to the Lebesgue integral, which isn’t an obscene time commitment.
Also, there’s some virtue to just being familiar with the definitions and concepts that everybody else is. (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?)
If you’re looking for an easier, more straightforward treatment than Rudin, I’d recommend Cohn’s Measure Theory. I’m not sure why, but it feels friendlier and less digressive.
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...
I’m not sure why you consider the gauge integral to be easier to understand the Lebesgue integral. It may be due to learning Lebesgue first, but I find it much more intuitive.
Also:
Royden is slightly better in this respect. The first four chapters are excellent, but still probably too theoretical. Further, eventually one will encounter measure spaces that aren’t based on the real numbers and the Lebesgue measure,
Yes, this is a good thing. One doesn’t understand a structure until one understands which parts of a structure are forcing which properties. Moreover, this supplies useful counterexamples that helps one understand what sort of things one necessarily will need to invoke if one wants certain results.
I’m not sure you made it to the end of that sentence. It is a good thing, but Royden obscures the connection with this four-chapter long digression into metric spaces and Banach spaces.
Maybe I misunderstand the needs of a budding Bayesian analyst. I agree that these sorts of counterexamples are useful, but perhaps not up front. I find it hard to imagine they’d encounter measure spaces that are either 1) not discrete or 2) not a subspace of R^n on a regular basis.
I haven’t studied real analysis, could you explain what advantages the guage integral is better than the lebesgue integral? Edit: maybe just respond to SarahC.
On (real) analysis: Bartle’s A Modern Theory of Integration.
Even Bayesian statistics (presumably the killer app for analysis in this crowd) is going to stumble over measure theory at some point. So this recommendation is made with that in mind.
The traditional textbooks for modern integration in this context are (the first chapters of) Rudin’s Real and Complex Analysis and (the first chapters of) Royden’s Real Analysis.
I can’t recommend Rudin because in the second chapter he goes on this ridiculously long tangent on Urysohn’s lemma that makes absolutely no sense to anyone who hasn’t seen topology before. Further, the exercises tend to have a difficulty curve that starts a bit too high for the non-mathically inclined.
Royden is slightly better in this respect. The first four chapters are excellent, but still probably too theoretical. Further, eventually one will encounter measure spaces that aren’t based on the real numbers and the Lebesgue measure, and because of the way Royden is set up the sections on Lebesgue theory and abstract measure theory are separated by a refresher on metric spaces and topology. Unlike the tangent in Rudin, this digression isn’t as avoidable.
My recommendation then corrects for these errors. Bartle’s book works with the gauge integral, which is perfectly compatible with the Lebesgue integral (i.e., they give the same results when both work) but has a more concrete formulation (not requiring any measure theory). I expected that a book taking this route would avoid measures altogether, but this is incorrect—even with the gauge integral questions of measurability come into play, and Bartle’s book covers these adequately.
As an aside, the gauge integral is one example of mathematicians failing to update, in a sense. It’s pretty superior to the Lebesgue integral in terms of conceptual simplicity and applicability, but practically no one uses it.
Hmm. Upvoted for contributing to a good topic but I’m not sure I agree.
I just looked up the gauge integral because I wasn’t familiar with it. For those curious about the debate, here’s the introduction to the gauge integral I found, which has a lot of relevant information. 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. The gauge integral approach doesn’t give you the tools to generalize to scenarios like Brownian motion where you need to construct different measures; also, the gauge integral doesn’t come with a lot of nice convergence theorems the way the Lebesgue measure does.
I don’t find the standard treatment of measure theory especially hard; it takes about a month to understand everything up to the Lebesgue integral, which isn’t an obscene time commitment.
Also, there’s some virtue to just being familiar with the definitions and concepts that everybody else is. (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?)
If you’re looking for an easier, more straightforward treatment than Rudin, I’d recommend Cohn’s Measure Theory. I’m not sure why, but it feels friendlier and less digressive.
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...
I’m not sure why you consider the gauge integral to be easier to understand the Lebesgue integral. It may be due to learning Lebesgue first, but I find it much more intuitive.
Also:
Yes, this is a good thing. One doesn’t understand a structure until one understands which parts of a structure are forcing which properties. Moreover, this supplies useful counterexamples that helps one understand what sort of things one necessarily will need to invoke if one wants certain results.
It is much easier to understand what the words in the definition of the gauge integral mean. It is harder to understand why they are there.
I’m not sure you made it to the end of that sentence. It is a good thing, but Royden obscures the connection with this four-chapter long digression into metric spaces and Banach spaces.
Maybe I misunderstand the needs of a budding Bayesian analyst. I agree that these sorts of counterexamples are useful, but perhaps not up front. I find it hard to imagine they’d encounter measure spaces that are either 1) not discrete or 2) not a subspace of R^n on a regular basis.
Yes, I did make it to the end of the sentence. But I misinterpreted the sentence to be having two distinct criticisms when there was only one.
I haven’t studied real analysis, could you explain what advantages the guage integral is better than the lebesgue integral? Edit: maybe just respond to SarahC.