2

It is well known that a function which is Riemann integrable is also Lebesgue integrable, and both integrations result in the same value.

Question: Can one give an example of a Riemann integrable function $f\colon [a,b]\rightarrow\mathbb{R}$, in which, computation of the integration by Lebesgue's approach is easier than the Riemann's approach?

(Of course, certain functions are Lebesgue integrable but not Riemann integrable, and computation of Lebesgue integration is also easy. I want an example to compare both the methods.)

Groups
  • 10,534
  • Can you define easier in this setting? – Siminore Jul 07 '15 at 11:17
  • Integral of a simple function (piecewise constant with finitely many discontinuities) follows immediately from the definition in the Lebesgue case. If you go via the Riemann route, approximating it with rectangles could get messy around the points of discontinuity. – Calculon Jul 07 '15 at 11:22

1 Answers1

1

The Thomae function is Riemann integrable, while the computation of its Riemann integral is harder than that of the Lebesgue integral: the function is zero almost everywhere, so its Lebesgue integral is zero.

Community
  • 1