What are the differences between Rieman integrl and Lebesgue integrl?

Question

I'd like to know

What are the differences between Rieman integral and Lebesgue integral?

especially,in real senses I mean if I have a bill in a restaurant and some coins in my pocket how do it (pay the bill) by means of Lebesgue integral and Riemann integral?

Thanks in advance

Answer 1

In reality, the two will totally coincide, in the sense that if you are integrating over some time interval $[a,b]$, and a function that is almost everywhere continuous (except on a set of measure zero,)

then the riemann integral will give the same evaluation as the lebesgue integral:

$$\int_{a}^{b} f(x)dx = \int_{[a,b]} f\, dm.$$

The Lebesgue integral extends the class of functions (they can be fairly discontinuous) one can integrate over, but also the sets that we care about! In other words, you don't need to integrate over an interval anymore, but you could instead integrate over almost any reasonable set (this is imprecise and sloppy.) For example, how can we make sense of $$\int_{[1,\infty)} \frac{1}{x^2} dx?$$

One answer is the Lebesgue integral. Even moreso, this is still a ray so it still can be made precise with improper integration. However, you can easily imagine that we don't want to always integrate over a connected component of the real line.

For your "real difference" criteria, there really isn't one:

"Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane." Quoted in N Rose Mathematical Maxims and Minims (Raleigh N C 1988).