Add additional types of measures early on

[qwinters]

https://github.com/Clopen-Analysis/latex/blob/c979e9e63487f1114e0e87cccbddf5d19e40f27a/chapters/ch1.tex#L732

Lebesgue measure is great and of course the standard, but I think it does make sense to give more examples of measures early on to encourage the reader to think about more than just what they already have immediate intuition for, and also to motivate why people should care about measure theory.

In particular I'm thinking of:

• The Hausdorff measure would be a good add - maybe biased here, but this turns up a lot in the areas I'm interested in and I think while the intuition is pretty clear here after you get the definition, it would still be worth while introducing early on
• An explicit construction of Haar measure on R^n would be good - in particular this one is actually pretty easy to construct (see e.g. Meckes - The Random Matrix Theory of the Classical Compact group which has 5 or 6 constructions all of which offer great intuition that eventually generalizes for the section on Haar measure/Harmonic analysis at the end)
• Dirac measure/point mass - In particular this is super important when dealing with things like Random matrix theory (e.g. the empirical spectral measure = linear combo of diracs at eigenvalues), and I think develops really nicely in parallel to indicator functions
• Gauss measure (in particular as a stand in for all probability measures. any first year grad student should be able to recognize this measure with probability 1, but it still is good for intuition)
• Circular angle measure on S^1 (and the connection to (n-1)-Hausdorff measure on S^{n-1}) - also gives some good intuitions and helps generalize polar integration techniques

These might also deserve to get put into the next chapter "More on Measures"

[abackus]

The Dirac measure appears in Chapter 1, and the circle measure appears in Chapter 2, but I agree that the other measures would be great examples to add too.

[qwinters]

Sorry must have missed those! Haven't had time to read fully in-depth, so mostly just skimming.

I can take on adding the Gaussian/Hausdorff/Haar measure for compact matrix groups in R^n in the near future (but a bit busy now so probably won't be until the weekend/sometime next week)

[abackus]

That would be great, thanks!

Also, writing this made me realize that the definition I gave of spherical measure doesn't actually work, so I'll fix that later today. It might be a good motivating example for Hausdorff measure (and Haar measure in case d = 1).