# Introducing the Lebesgue integral before Riemann's

Has anyone attempted to introduce, or has data on such endeavor, Lebesgue integration before Riemann? I've seen many discussions about how the Riemann integral is obsolete and that it is presented because it appeals to intuition*, contrary to Lebesgue's theory, which is considerably harder.

As an aside, it does not have to be Lebesgue's integration theory exclusively. I've seen people advocating for the gauge integral as well, but I don't know much.

*Citation needed. I don't think I know anyone that understood the geometric appeal of the Riemann integral when meeting it for the first time.

• (Very) related: mathoverflow.net/q/52708/22971 Jun 1, 2014 at 1:47
• I "got" the geometric interpretation of Riemann's integral the first time. But then again, I doubt we have ever met ;-) Jun 1, 2014 at 2:04
• If you're interested in using more powerful integrals, introduce the Henstock integral. It's conceptually easier than the Lebesgue integral and just as powerful. Jun 1, 2014 at 2:13
• @nomen: That's certainly true, and part of me thinks it would be neat to teach the Kurzweil-Henstock integral in undergraduate analysis. On the other hand, the "conceptually harder" stuff behind the Lebesgue integral is the theory of measure. This hard material is much more ubiquitous in higher mathematics than the ability to integrate highly oscillatory highly discontinuous highly unbounded functions defined on the real line. So I would be careful in short-changing the Lebesgue integral from the graduate analysis syllabus. Jun 1, 2014 at 2:30
• @Pete: That is a good point. My suggestion comes from an "economic" argument. The undergrad can get all the power of the Henstock integral with just marginally more effort than the Riemann integral. But measure theory is unmotivated, until it is motivated. Perhaps that is why Riemann is still around. Jun 1, 2014 at 2:33

In the US, you'd be hard-pressed to find any student seeing the Lebesgue integral before having ever seen the Riemann integral. Every calculus book I've seen defines the integral as the Riemann integral. That said, when I was a graduate student, I came across this book

Lebesgue Integration and Measure by Alan J. Weir

that, instead of connecting the Lebesgue integral to the Riemann integral, connects it directly to the problem of finding areas. This might not be what you want, but it's the only textbook with the Lebesgue integral that I know of that is both written for undergraduates and doesn't have a chapter or section explicitly on the Riemann integral.

A quick internet search tells me the math department at the University of York uses this text and in its standard sequence doesn't appear to make reference to the Riemann integral. You might consider contacting someone there.

• Surely the Riemann integral appears in a prerequisite for this course. At least one version of third year Lebesgue Integration at UofY (maths.york.ac.uk/www/node/13997) begins its Aims section with: "The Lebesgue integral is an extension of the Riemann integral studied in Real Analysis, able to integrate many more functions." Jun 1, 2014 at 11:31
• I thought the same thing. But then I looked at the chain of prerequisites and didn't see anything about the Riemann integral. It might be that it's assumed that the students have seen in school.
– ncr
Jun 1, 2014 at 12:18
• Hilary Priestley's undergrad textbook Introduction to Integration ukcatalogue.oup.com/product/9780198501237.do has both a stripped-down and full version of the Lebesgue integral. It's a really excellent, clear book. Jun 1, 2014 at 12:29

I believe you answer youself: Riemann's integral (while intuitively quite appealing) isn't easy to get to grip with. Lebesgue's definition is much harder to aprehend, and solves problems with Riemann's integral that have to be explained it that terms. So you would be introducing a more complex definition, while taking out the justification to define it. For "everyday engineering use" Riemann's integral is more than enough, most of the times integrals of interest are expressible through antiderivatives.

• I didn't ask whether to introduce Lebesgue or Riemann first, but if someone has tried to do the former and what were the learning outcomes. Jun 1, 2014 at 2:21