Math 471 Real Analysis I, Fall 2025



Instructor: Alex Iosevich
Meeting Time: MW 10:25-11:40 a.m.
Textbook: Measure, Integration and Real Analysis, by S. Axler (link to Axler's website)
Office Hours: MW 9:00-10:00 a.m.
Grading: Homework (%30), Midterm (%30), and Final (%40)

Preparing for this class:

This is a graduate level course in measure theory. In order to get the most out of this course, I recommend reviewing undergraduate analysis, linear algebra, and, just as importantly, practicing applying these concepts in a direct and elementary way. Among the elementary concepts I recommend reviewing is basic inequalities such as the Cauchy-Schwarz inequality, Young's inequality, and Jensen's inequality.

To give you an illustration of what I mean by applying those inequalities, consider the following problem. Consider an n by n matrix AA with 1-0 entries entries {aij}1≤i,j≤n{\{a_{ij} \}}_{1 \leq i,j \leq n}   satisfying aijaij′=1a_{ij} a_{ij'}=1  for at most one value of i if j¬=j′.j \not=j'. What is the largest possible number of 1's as a function of n in such a matrix, up to a constant? Can you construct a matrix that realizes this bound? Can you construct a geometric problem where this calculation naturally comes up?

One of my main goals when I teach is to bridge the gap between classwork and research. When you work on a research problem, it typically does not come with a sign in neon lights announcing which techniques from your graduate courses you need to use in order to solve it. Furthermore, even if you identify the right technique, you typically need to perform quite a bit of technical work to execute the idea. This type of bridging will be emphasized throughout the course.

Here is another example of what I mean by reviewing elementary concepts. The following problem can be found in Edmund Landau's calculus book as an exercise. Let f∊C1(a,b),f \in C^1(a,b), and suppose that f′(x)≥1.f'(x) \ge 1. Further suppose that f′f'       is monotonic. Can you prove that |∫abeiRf(x)dx|≤4|R|?\left| \int_a^b e^{iRf(x)} dx \right| \leq \frac{4}{|R|}? (a
Can you further show that if f is twice continuously differentiable on [a,b][a,b]  , and f″(x)≥1f''(x) \ge 1   , then
|∫abeiRf(x)dx|≤C′|R|?\left| \int_a^b e^{iRf(x)} dx \right| \leq \frac{C'}{\sqrt{|R|}}?
Another great problem to review is the following. Can you determine all complex values z, with |z|=1,|z|=1, such that
∑n=1∞znn\sum_{n=1}^{\infty} \frac{z^n}{n} converges conditionally?

Another aspect of analysis I would like to emphasize, to the extent the time and curriculum requirements allow, is applications to related areas of mathematics. Here is an example of a fun problem to think about, if you have not seen it before. Consider Ω,\Omega, a bounded closed region in d-dimensional Euclidean space of area k,k, a positive integer. Can prove that in a "typical" translate Ω+τ\Omega+\tau  , where the notion of "typical" is left for you to nail down the number of  of integer lattice points is ≤k?

Here is another elementary example worth recalling. Suppose that C is a closed twice differentiable convex curve in the plane parameterized by ∫01k(t)dt=2π?\int_0^1 k(t) dt= 2\pi ? Now redefine the curvature function as follows. Let N(x) denote the function from C to the unit circle where every point x on C is taken to the outward unit normal at x. Prove that k(t)=det(dN)x, where t is the parameter corresponding to the point x on C in the parameterization of C given above. k(t)=det(dN)_x,
\leq k?