In which way Lebesgue Integral integrates over values?

Question

A special tutorial for full dummies says

In order to distinguish between the Lebesgue and Reimann integrals consider the values that the function f can take to be on the x-axis (called the domain of the function) and the values of the function evaluated at the chosen points to be on the y-axis (called the range of the function). The Lebesgue integral defines the sub-intervals along the range of the function and the Riemann integral defines them along the domain of the function.

A thorough specification, introducing the Lebesgue integral in terms of sequences of countable sets and measures, precedes this conclusion. I am lost in all of these measure spaces. I do not see how integration over y axis follows from them. Can you draw the parallels, probably giving some examples? I feel like Lebesgue integral lacks examples.

Answer 1

For a nonnegative function $f$, you can introduce a family of sets $\{ \{ A_{k,n} \}_{k=0}^{n^2} \}_{n=1}^\infty$, where $A_{k,n}=\{ x : k/n \leq f(x) < (k+1)/n \}$ for $k<n^2$ and $A_{n^2,n}=\{ x : f(x) \geq n \}$. This is a partition of the domain, which is given by dividing up the range into intervals and taking the preimage of each of these intervals under $f$. If $f$ is measurable, then the sets $A_{k,n}$ are all measurable. Thus you have a sequences of numbers $I_n = \sum_{k=0}^{n^2} \frac{k}{n} m(A_{k,n})$. Each $I_n$ is the integral of a "simple function", which is equal to $k/n$ on $A_{k,n}$.

It turns out that $\lim_{n \to \infty} I_n = \int f dm$. So while the Riemann integral approximates functions by functions which are constant on intervals, the Lebesgue integral approximates functions by functions which are constant on general measurable sets.

Similar to the Riemann case, actually computing an integral this way is usually hard. One case where it can be done is when $f$ is increasing: in this case the sum above is the left Riemann sum of $f$, albeit in general over a non-uniform partition. You might try writing out the calculation of $\int_0^1 x^2 dx$ by this method for instance.

Answer 2

If you draw a picture of function f (for clarity, single varibale) with x as horizontal axis. You have two ways to slice area enclosed by f from x = a to x = b, and y = 0 with infinitimal rectangular bars, and sum them. Riemann integral said "let us slice them into vertical infinitesmal rectangular bars and sum the area of vertical infinitesmal rectangular bars. Lebesgue said how about we slice them with horizontal infinitesmal bars and sum over horizontal infinitesmal bars?

It turns out that if the summation in Riemann way converges, you can do it in Lebesgue way and get the same value, and not other way around.

In plain language, Lebesgue way solves a problem that a single value of f might have positive measure of the domain corresponding to it that Riemann way misses.