Integration – Riemann vs. Lebesgue Integration: A Comparative Analysis

Question

If a function is not Riemann integrable, what does it mean geometrically?

Is it that we can't use integration to find out area under the curve?
So basically when a function is Riemann integrable then it can tell us about area.

I am asking this because I saw a function which is not Riemann integrable but Lebesgue integrable.
What does it mean geometrically?

Answer 1

Usually non-Riemann integrable functions are pathological functions like the Dirichlet function etc. But these are Lebesgue integrable.

One of the main implications of a function being non-Riemann integrable is that it is not continuous enough to have a well-defined area under the curve. But the mathematical definition limits itself to the existence of a limiting sum only. In that sense, it is quite rigorous.

I further quote from this article:

We realize that both of them can help us to integrate functions. The difference is that the Riemann integral subdivides the domain of a function, while the Lebesgue integral subdivides the range of that function. The step function for Riemann integral has a constant value in each of the subintervals of the partition, while the simple function for Lebesgue integral provides finitely many measurable sets corresponding to each value of that function. The improvement from the Riemann integral to the Lebesgue integral is that the Lebesgue integral provides more generality than the Riemann integral does. From the reverse perspective, the Riemann integral can imply the Lebesgue integral.

Hope this helps.