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
Lesbegue integral can be seen as an extension of the Reimann integral. Whenever you can use Reimann integral you can also use the Lesbegue integral, and the results would be the same. We use Reimann in these cases because it’s simpler. However, the Lesbegue integral can also be used in cases where the Reimann integral can’t be applied.
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.
Lebesgue integrals and Riemann integrals are two different things that both take functions as input and give you some real number which corresponds in some sense to an area (or more general to a volume if you go to higher dimensions).
1) Historically the Riemann integral existed first and is in most cases the weaker of the two, lebesgue used a more general way of measuring volumes. Lebesgue integrals can be used on much more abstract spaces, where as Riemann integrals are only defined on the real numbers (and vector spaces on real numbers, which includes the complex numbers).
2) Not every function that can be riemann integrated can be lebesgue integrated, any function with both infinite area above and below the x-axis can’t be lebesgue-integrated, even if the riemann integral would be finite (because of positive and negative area cancelling to a finite value). But that’s also the only case where problems occure.
On the other hand most lebesgue-integrable functions can not be riemann-integrated as Riemann needs a function that is “mostly” continous.
Where both the riemann and the lebesgue integral exist and are finite, they also coincide.
3) Integrals in general are used everywhere in physics, so that’s why. Why both? Given a certain problem, a riemann integral is usually much easier to compute. Also many important theorems (for example that derivatives and integrals in some sense cancel each other) are derived directly from riemann integrals. On the other hand you can prove other really powerful theorems with lebesgue that hold for all lebesgue integrals quite easily. And as the values of the two integrals basically always coincide, this allows you to chose which definition to use based on your use-case.
Also as mentioned before lebesgue integrals are something much more general. They are part of a branch of maths called measure theory, which is the mathematical way of measuring volume in abstract spaces. This for example is the basis of stochastics. You could say lebesgue integrals are “just” a side effect of a much more general theory used in many other parts of maths and physics.
This also means that lebesgue integrals are much harder to understand, I don’t think you’d want to teach measure theory before knowing what the idea of an integral is.
4) Not sure what exactly you want to hear here. You can define the riemann Integrals on higher dimensional spaces (you can associate the complex numbers with a real numbered plane R^2), but this get’s quite ugly I’ve been told. lebesgue integrals on the other hand work just the same on higher dimensional spaces, this is quite pretty. But as soon as you want to actually compute such an integral you usually split it up into multiple 1-dimensional integrals and use the riemann integral again, as this is usually the easiest way.
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