Milind Hegde

Power Series. Definition of outer measure and Lebesgue measure. Measurable sets and measurable functions. Step functions and simple functions. Egorov Theorem and Lusin Theorem. Fubini Theorem and Tonelli Theorem. Fourier Series and Transforms. Multivariable Analysis.

We covered measure theory on R^n—we constructed the Lesbegue measure and integral, and proved facts about the Lesbegue integral like the dominated convergence theorem and Fatou's lemma. After this, we discussed the basic theory of Hilbert and Banach spaces (including a class on the L^2 convergence of Fourier series), before moving onto multivariate calculus, proving the chain rule, and thereoms like Clairaut's, inverse function, and implicit function. Note that most classes do multivariate calculus before measure theory.

Measure theory, Lebesgue Integration, Hilbert Space, Fourier Series, Multivariable Analysis

A lot about measure theory. And analysis proof writing skills.

My neutral choice of recommendation is that this course is probably too difficult and unnecessary for students do not want to pursue a graduate degree in math. But it is a great class for those who want to study probability theory or applied math.

Good class, but could be unclear at times, and lacks supporting materials (no posted class notes, study guides, written past homework solutions). If you fall behind in class, it’s pretty hard to catch up, and there was not a sole textbook the class referred to.

real analysis, multivariate analysis, elements of measure theory and functional analysis

Measure Theory, Functional Analysis, and Multivariable Analysis

Just look at the syllabus, more stuff on sequences and series in rudin, stein and shakarchi measure theory, then derivatives in higher dimensions with terry tao's book.

1 week: Review of Analysis I (Uniform convergence of power series, Log and Exp, Cantor function) 8 weeks: Measure Theory (Outer measure, Lebesgue/Borel measurable sets, Lebesgue Integration, Limit theorems - BCT, MCT, DCT, Fubini) 1 week: Hilbert spaces, Fourier basis 3 weeks: Multivariable analysis (chain rule, inverse function theorem, implicit function theorem)