## Little did I know (real analysis edition)

I just finished teaching Math 521, undergraduate real analysis. I first took on this course as an emergency pandemic replacement, and boy did I not know how much I would like teaching it! You get a variety of students — our second and third year math majors, econ majors aiming for top Ph.D. programs, financial math people, CS people — students learning analysis for all kinds of reasons.

A fun thing about teaching outside my research area is encountering weird little facts I don’t know at all — facts which, were they of equal importance and obscurity and size and about algebra, I imagine I would just know. For instance, I was talking about the strategy of the Riemann integral, before launching into the formal definition, as “you are trying to find a sequence of step functions which are getting closer and closer to f, because step functions are the ones you a priori know how to integrate.” But do Riemann-integrable functions actually have sequences of step functions converging to them uniformly? No! It turns out the class of functions which are uniform limits of step functions is called the regulated functions and an equivalent characterization of regulated functions is that the right and left limits f(x+) and f(x-) exist for any x.

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## 11 thoughts on “Little did I know (real analysis edition)”

1. Jimmy Vineyard says:

It’s been great having you as a professor!

2. C.P. says:

I found exactly the same thing when I taught a two-semester undergrad real analysis course. Two gems I came away with: (1) a realization of just how discordant integral convergence and pointwise convergence can be (e.g. there is a sequence of nonnegative, uniformly bounded functions which converge in integral to zero but do not converge pointwise anywhere) and (2) sufficient conditions for the linear span of a sequence of monomials to be dense in C[a,b], the continuous functions on [a,b] with the sup norm (while discussing the Weierstrass approximation theorem, a student noticed that in one of these examples, all of the polynomials I constructed happened to be even and asked how sparse the approximating polynomials could possibly be which led me to read about the Muntz-Szasz theorem.)

3. George J. Ross says:

It would be nice if all Riemann-integrable functions were uniform limits of step functions; unfortunately, it ain’t so. Example: function f(x)=sin(1/x) on (0,1), f(0)=0 is Riemann-integrable but it cannot be uniformly step-approximated (troubles are at 0).

4. John Franks says:

I first encountered the concepts of regulated functions and the regulated integral in the text {\it Foundations of Modern Analysis} by Jean Dieudonne. I found them sufficiently enticing that I taught a freshman honors calculus course using the introductory text by Phillip Curtis which develops integration via regulated functions and the regulated integral rather than the Riemann integral. The appeal is the centrality of step functions for which the definition of the integral and its computation are trivial.

Of course one can (and to my mind should) define Riemann integrability using step functions: A bounded function f:[a,b] -> R is Riemann integrable if and only if for every epsilon > 0 there are step functions u and v such that u(x) \le f(x) \le v(x) for all x and \int v(x) dx – \int u(x) dx R the following are equivalent:
(1) f is measurable, (2) f is Lebesgue integrable, (3) f is the uniform limit of simple functions, and (4) for every epsilon > 0 there are simple functions u and v such that u(x) \le f(x) \le v(x) for all x and \int v(x) dx – \int u(x) dx < epsilon.

This is a straightforward result easily accessible to undergraduates once Lebesgue measure has been introduced.

5. John Franks says:

Somehow in the cutting and pasting my previous post got garbled. Here is a corrected version.
I first encountered the concepts of regulated functions and the regulated integral in the text {\it Foundations of Modern Analysis} by Jean Dieudonne. I found them sufficiently enticing that I taught a freshman honors calculus course using the introductory text by Phillip Curtis which develops integration via regulated functions and the regulated integral rather than the Riemann integral. The appeal is the centrality of step functions for which the definition of the integral and its computation are obvious.

Of course one can (and to my mind should) define Riemann integrability using step functions: A bounded function f:[a,b] -> R is Riemann integrable if and only if for every epsilon > 0 there are step functions u and v such that u(x) \le f(x) \le v(x) for all x and \int v(x) dx – \int u(x) dx R the following are equivalent:
(1) f is measurable, (2) f is Lebesgue integrable, (3) f is the uniform limit of simple functions, and (4) for every epsilon > 0 there are simple functions u and v such that u(x) \le f(x) \le v(x) for all x and \int v(x) dx – \int u(x) dx < epsilon.

This is a straightforward result easily accessible to undergraduates once Lebesgue measure has been introduced.

6. JSE says:

Full version of John’s comment:

“I first encountered the concepts of regulated functions and the regulated integral in the text
{\it Foundations of Modern Analysis} by Jean Dieudonne. I found them sufficiently enticing that I taught a freshman honors calculus course using the introductory text by Phillip Curtis which develops integration via regulated functions and the regulated integral rather than the Riemann integral. The appeal is the centrality of step functions for which the definition of the integral and its computation are obvious.

Of course one can (and to my mind should) define Riemann integrability using step functions: A bounded function f:[a,b] -> R is Riemann integrable if and only if for every epsilon > 0 there are step functions u and v such that u(x) \le f(x) \le v(x) for all x and \int v(x) dx – \int u(x) dx R the following are equivalent:
(1) f is measurable, (2) f is Lebesgue integrable, (3) f is the uniform limit of simple functions, and (4) for every epsilon > 0 there are simple functions u and v such that u(x) \le f(x) \le v(x) for all x and \int v(x) dx – \int u(x) dx < epsilon.

This is a straightforward result easily accessible to undergraduates once Lebesgue measure has been introduced."

7. George J. Ross says:

I happen to have Dieudonne’s “Foundations” on my bookshelf. On page 132, the author writes: “The last section of Chapter VII introduces, as a useful technical tool in the development of calculus, a category of functions which are classically described as “functions with discontinuities of the first kind”; in an effort toward a more concise expression, and to avoid one more use of the overworked term “regular”, the author has tentatively introduced the neologism “regulated functions” (corresponding to the French “ Fonction réglée”), which he hopes will not sound too barbaric to English-speaking readers.”

8. George J. Ross says:

Undergraduate Analysis at Warwick (sophomore level):
https://warwick.ac.uk/fac/sci/maths/people/staff/oleg_zaboronski/analysisiii/
based on Walker’s:
Chapter 4:
4. Constructive Integration……………………………… 103
4.1 Step Functions ……………………………………. 104
4.2 The Integral of a Regulated Function ………………….. 107
4.3 Integration and Differentiation ……………………….. 111
4.4 Applications ……………………………………… 117
4.5 Further Mean Value Theorems ……………………….. 119
Exercises ……………………

9. George J. Ross says:

There are five undergraduate level books (each introducing the Regulated Integral concept) written by five Bourbaki members:
(1) Fonctions d’une variable réelle by Nicolas Bourbaki
(2) Foundations by Jean Dieudonné
(3) Cours d’analyse by Laurent Schwartz
(4) Undergraduate Analysis by Serge Lang
(5) Mathématiques générales by Charles Pisot and Marc Zamansky (the text is accompanied by Algebra and Analysis Problems and Solutions by G. LeFort)
The last of the five books (the one by Pisot and Zamansky) used to be extremely popular in Europe the 1960’s and 1970’s as a freshman text for math majors.

10. George J. Ross says:

Roger Godement‘s Analysis IV on pages 1 through 4 contains a discussion of the Bourbaki vs Lebesgue approaches to integration. Godement begins with: “Bourbaki has sometimes been blamed for giving priority to function integrals instead of first defining measures on sets like everyone and like Borel and Lebesgue”. Godement’s train of thought is interesting …