I had just learned in measure theory class about the monotone convergence theorem in this version:

For every monotonically increasing sequence of functions $f_n$ from measurable space $X$ to $[0, \infty]$, $$ \text{if}\quad \lim_{n\to \infty}f_n = f, \quad\text{then}\quad \lim_{n\to \infty}\int f_n \, \mathrm{d}\mu = \int f \,\mathrm{d}\mu . $$

I tried to find out why this theorem apply only for a Lebesgue integral, but I didn't find a counter example for Riemann integrals, so I would appreciate your help.

(I guess that $f$ might not be integrable in some cases, but I want a concrete example.)


3 Answers 3


Riemann integrable functions (on a compact interval) are also Lebesgue integrable and the two integrals coincide. So the theorem is surely valid for Riemann integrals also.

However the pointwise increasing limit of a sequence of Riemann integrable functions need not be Riemann integrable. Let $(r_n)$ be an ennumeration of the rationals in $[0,1]$, and let $f_n$ be as follows:

$$f_n(x) = \begin{cases} 1 & \text{if $x \in \{ r_0, r_1, \dots, r_{n-1} \}$} \\ 0 & \text{if $x \in \{ r_n, r_{n+1}, \dots \}$} \\ 0 & \text{if $x$ is irrational} \\ \end{cases}$$

Then the limit function is nowhere continuous, hence not Riemann integrable.

  • 3
    $\begingroup$ I think the problem is more the other way round. If you start with a sequence of Riemann integrable functions $(f_n)_{n\in\mathbb{N}}$ that converges point-wise to $f$. Is $f$ again Riemann integrable? And what about the szenario where you are not on a compact interval? $\endgroup$ Nov 29, 2019 at 9:53
  • $\begingroup$ @NathanaelSkrepek Thank you. I have given a counter-example for the domain $[0,1]$. $\endgroup$ Nov 29, 2019 at 10:02

Here is a version of the Monotone convergence theorem for Riemann integrals that can be proved without referring to measure theory:

Theorem. Let $\{f_n\}$ be a nondecreasing sequence of Riemann integrable functions on $[a,b]$ converging pointwise to a Riemann integrable function $f$ on $[a,b]$. Then $$ \lim_{n\to \infty}\int_a^b f_n(x)\,dx=\int_a^b f(x)\,dx. $$

An elementary proof is given in this paper.

  • 1
    $\begingroup$ thanks! so I guess that I'm looking for an concrete example of monotonic series of positive riemann integrable functions whose limit is not riemann integrable. $\endgroup$ Nov 29, 2019 at 9:59
  • 1
    $\begingroup$ @AmitKeinan I have given such an example in my answer. $\endgroup$ Nov 29, 2019 at 10:02
  • 2
    $\begingroup$ Look at this question. $\endgroup$
    – user140541
    Nov 29, 2019 at 10:03

I spent about two weeks working on an elementary proof of this problem earlier this year.

There is an elementary proof of something slightly stronger: Let $f_n$ tend pointwise to $0$ on $[0,1]$, with $f_n$ Riemann integrable, and for all $x$ sup $|f_n(x)| \leq 1$. (This has the obvious generalisation to an arbitrary interval, with convergence pointwise to a Riemann integrable function and $f_n$ uniformly bounded).

I would be happy to write up my proof after my exams are finished this summer :)

This is a slightly harder version of a problem given on this problem set for second year undergraduates at the university of cambridge, although that also allows the assumption that the $f_n$ are continuous.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.