My dad always says the best way to tell a good undergraduate math program is to see if they offer Lebesgue (not correct spelling, I KNOW) Theorem to Math Undergraduate, any other tells??
The Princeton Review offers a sampling of strong math programs in one of its categories, “Great Schools for Mathematics Majors.” These are a few of the 26 colleges included:
Bowdoin
Caltech
Carleton
Hamilton
Harvard
Harvey Mudd
Haverford
MIT
Rice
UChicago
You may also want to look at [a school’s performance in the Putnam Competition](The Putnam Archive).
You also could look at the number of alumni-earned PhDs in math & statistics, which you can find here:
https://ncsesdata.nsf.gov/webcaspar/
By absolute number of math/stat PhDs earned from 2010-2014 (unadjusted for school or program size), here are the top 20:
94 University of California, Berkeley
67 University of Chicago
65 Harvard University
59 Massachusetts Institute of Technology
49 California Institute of Technology
43 Brigham Young University, Provo
43 Cornell University
42 University of California, Los Angeles
39 Princeton University
39 University of Michigan, Ann Arbor
38 Stanford University
36 Harvey Mudd College
35 University of Florida
33 Columbia University in the City of New York
33 University of Texas at Austin, The
32 North Carolina State University
31 University of California, San Diego
30 Carnegie Mellon University
29 University of Maryland, College Park
28 St. Olaf College
These schools differ greatly in size (from Harvey Mudd with ~800 students to UT-Austin with nearly 40K undergrads).
We can derive per capita rates by using IPEDS data for the number of bachelor degrees conferred in math/stat. Using 2009 as the reference year, the following appear to be some of the top schools by percentage of math/stat majors who go on to earn PhDs in math/statistics.
[QUOTE=""]
30%
… Harvey Mudd
… Caltech
[/QUOTE]
21%-30%
… Cornell
… Swarthmore
… Williams
… Oberlin
… UChicago
… Princeton
… Georgia Tech
… Reed
… Stanford
11%-20%
… Carleton
… St. Olaf
… Rice
… Harvard
… Duke
… WUSTL
… BYU
… Carnegie Mellon
… MIT
… UPenn
… NYU
… Columbia
Schools with 6%-10% rates include:
Berkeley, Dartmouth, UFlorida, Yale, UCLA, Brown, UMichigan, Northwestern, Vanderbilt
Caveats: these numbers may be confounded by factors other than math program quality, such as admission selectivity or alumni career preferences. The order within groups probably isn’t too significant. If one chose a population reference year other than 2009, the results might be a little different. The per capita list may not be exhaustive (because I may have overlooked a few high-performing schools and neglected to calculate their rates).
Most of the universities with rates above 5% also show up among the USNWR top ~40 mathematics graduate programs.
http://grad-schools.usnews.rankingsandreviews.com/best-graduate-schools/top-science-schools/mathematics-rankings
For your specific question, you may want to look at course descriptions in the catalogs and math department web pages, though sometimes it may just say “measure theory” instead of “Lebesgue”. The usual courses covering these topics would be second course in real analysis or graduate level analysis (which undergraduates sometimes take).