inb4 “not eli5-able”
As the rules say, the replies should be aimed at laymen, not five year olds. I let you assume the laymen reading this have at least had a calculus course (so you don’t have to explain how integral is are under the curve), maybe even heard of some basic real analysis like the epsilon-delta definition, but whose knowledge in any of this is rudimentary.
Roughly these questions which I think are mostly connected:
1. Why do we have these different definitions of integrals in the
first place
2. When are they not equivalent?
3. Why would anyone care about them, study them?
4. Assuming these answers are mostly concerned with R, in the complex world does anything change?
In: 4
For many integrals, they are exactly the same, but there are some awkward functions that are Lebesgue integrable but not Riemann integrable, one simple example being the Dirichlet function, which takes the value 1 for rational numbers and 0 for irrational numbers. Obviously that’s a bit of a contrived example, but it’s hard to think of something straightforward where the distinction has an important impact. I remember using Lebesgue integrals a lot when studying numerical analysis of differential equations, because in that field you’re often working with slightly degenerate functions. For example, you might be trying to approximate the solution to a PDE by a piecewise linear function (one made up of lots of segments of straight lines/flat planes/hyperplanes). The obvious way to measure how close the approximation is is by plugging it into the equation to find out how far off it is, and then integrating that over the relevant domain. However, you run into the problem that the approximation doesn’t have well-defined derivatives at the boundaries between the linear segments. You can use Lebesgue integration to get around this problem.
Latest Answers