I wanted to take measure theory in college, but my advisor talked me out of it, saying that it is an old, ossified field where writers play games of streamlining their proofs. They seek too much generality and defer applications to later courses. That complaint could apply more generally, that introductory graduate classes are bad because they have captive audiences, but it seems to me much worse in analysis than other fields of mathematics. What is the point of measure theory? Archimedes gave a rigorous delta-epsilon proof that if there is a coherent notion of measure, then the area of a circle is πr². But how do you know that you don’t encounter inconsistencies?
Applications are related to constructibility. If you know what your goal is, you can see if you can skip the axiom of choice. Indeed, as I phrased it above, the goal is to show that measure is defined on some sigma algebra, not just the maximal one. And it is also related to constructibility. Why do we want measureable functions? What is a function? If a function is something you can apply at a point, then from a constructive viewpoint it must be continuous. But you can constructively describe things like infinite Fourier series. You can’t evaluate them at points, but only do other things, like compute an average over a small interval. You want a theorem that the Hilbert space of square integrable functions on the circle is isometric to the Hilbert space of square summable sequences L²(S¹)=ℓ². Usually you define L² as measurable functions up to the equivalence relation of equality away from a measure zero set. But you could instead define it as the metric completion of infinitely differentiable functions under the appropriate norm. This is a much better definition for many reasons, including constructibility, but it requires you to open up your definition of function.
Here are two alternate books. Measure and Probability by Adams and Guillemin is a book about measure theory that tries to justify it by the context of things like 0-1 laws of probability. I’m not sure it succeeds in the justification, but it gives something more serious to think about if you want to drop the axiom of choice or the law of the excluded middle. Also, see this MO question.
The second book is more advanced, outside of the scope of this post. After measure theory, one has functional analysis, the study of infinite dimensional topological vector spaces of functions. I once heard it described as “degenerate topology.” For this, I recommend Essential Results of Functional Analysis by Robert Zimmer. It gives a bunch of applications to differential equations with a geometric flavor. It minimizes the amount of theory to get to the applications, in particular, by only using Hilbert spaces, not general Banach spaces.
I wanted to take measure theory in college, but my advisor talked me out of it, saying that it is an old, ossified field where writers play games of streamlining their proofs. They seek too much generality and defer applications to later courses. That complaint could apply more generally, that introductory graduate classes are bad because they have captive audiences, but it seems to me much worse in analysis than other fields of mathematics. What is the point of measure theory? Archimedes gave a rigorous delta-epsilon proof that if there is a coherent notion of measure, then the area of a circle is πr². But how do you know that you don’t encounter inconsistencies?
Applications are related to constructibility. If you know what your goal is, you can see if you can skip the axiom of choice. Indeed, as I phrased it above, the goal is to show that measure is defined on some sigma algebra, not just the maximal one. And it is also related to constructibility. Why do we want measureable functions? What is a function? If a function is something you can apply at a point, then from a constructive viewpoint it must be continuous. But you can constructively describe things like infinite Fourier series. You can’t evaluate them at points, but only do other things, like compute an average over a small interval. You want a theorem that the Hilbert space of square integrable functions on the circle is isometric to the Hilbert space of square summable sequences L²(S¹)=ℓ². Usually you define L² as measurable functions up to the equivalence relation of equality away from a measure zero set. But you could instead define it as the metric completion of infinitely differentiable functions under the appropriate norm. This is a much better definition for many reasons, including constructibility, but it requires you to open up your definition of function.
Here are two alternate books. Measure and Probability by Adams and Guillemin is a book about measure theory that tries to justify it by the context of things like 0-1 laws of probability. I’m not sure it succeeds in the justification, but it gives something more serious to think about if you want to drop the axiom of choice or the law of the excluded middle. Also, see this MO question.
The second book is more advanced, outside of the scope of this post. After measure theory, one has functional analysis, the study of infinite dimensional topological vector spaces of functions. I once heard it described as “degenerate topology.” For this, I recommend Essential Results of Functional Analysis by Robert Zimmer. It gives a bunch of applications to differential equations with a geometric flavor. It minimizes the amount of theory to get to the applications, in particular, by only using Hilbert spaces, not general Banach spaces.