Lebesgue Integration Formula Help

Question

I have been endeavouring to teach myself Lebesgue's method of integration, and while theory has been okay, I have had some severe difficulty with the practical side (most texts seem to not provide a lot of basic examples other than characteristic functions), and hope someone can help me. I understand that the formula for the integral is $\int_{a}^{b} f(x) = \lim_{n\to\infty} \sum\limits_{i=1}^{n} (y_i')[(f^{-1}(y_{i+1}))-(f^{-1}(y_{i}))]$, where the inverse function subtraction should be the measure of the interval. I am having trouble selecting $y_i'$. In the analogous Riemann integral, $(x_i')$ is simply $a+i[(b-a)/n]$, but I can't figure out how to do the same on the range. If someone could provide one or two worked examples of integrals by the formula (like of $\sqrt{x}$), with limits evaluated, it would be of great help. Thank you in advance!

Answer 1

I think that is somewhat of an overly complicated way of thinking about it, and also locks you into the monotone case, where the Lebesgue theory doesn't get you any new integrable functions.

Let's deal just with nonnegative functions for now. Partition the range of a nonnegative, possibly unbounded function $f$ into intervals as follows:

$$P_n = \{ [0,1/n],[1/n,2/n],\dots,[n-1/n,n],[n,\infty) \}.$$

Let these intervals be denoted by $\{ J_k \}_{k=0}^{n^2}$, and their inverse images under $f$ be denoted by $\{ I_k \}_{k=0}^{n^2}$. (I need $n^2$ because I'm increasing the upper bound to $n$ and simultaneously improving the resolution to within $1/n$.) Then an approximation for the integral is given by

$$\sum_{k=0}^{n^2} \frac{k}{n} m(I_k).$$

There is nothing special about the choice $\frac{k}{n}$; for each $k=0,\dots,n^2-1$ I can choose any number in $\left [ \frac{k}{n},\frac{k+1}{n} \right ]$, and for $k=n^2$ I can choose any number in $[n,\infty)$.

One thing that may help is that in the case of a continuous function, the Lebesgue integral is actually being constructed in exactly the same way as the Riemann integral. This is especially easy to see in the case of a monotone function.

For instance, when you integrate $\sqrt{x}$ on $[0,4]$ using the method I just described, you are chopping up the range $[0,2]$ into pieces $J_k$, finding the intervals $I_k$ that map into the $J_k$, and then adding up $\frac{k}{n} m(I_k)$. But since $I_k$ are just intervals and $\sqrt{x}$ is increasing, this is exactly the left Riemann sum for the corresponding Riemann integral on a particular partition.

Answer 2

First it is worth noting that any Riemann integrable function on Euclidean space is also Lebesgue integrable. The intuition behind this proof is that the class of step functions, which are used to define Riemann integration, is a subset of the class of all linear combinations of characteristic functions (often referred to as simple functions).

That being said, the Lebesgue integral of Riemann integrable functions like $\sqrt x$ can be computed just the same way as the Riemann integration. (e.g. trapezoidal rule)

I believe a more interesting case would be to obtain Lebesgue integral of non-Riemann integrable functions, such as $f\left(x\right)=\bf{1}_{x\in Q\cap\left[0,1\right]}$ where $Q$ is the set of all rational numbers. First few pages of the following link show explicitly how to compute Lebesgue integral in these cases.

http://www.math.washington.edu/~hart/m555/integration.pdf

I think the underlying assumption for $f$ in your question was strict monotonicity, since you use the inverse of $f$. However, again, the class of all strictly monotonic functions on an interval is in the Riemann integrable function class. (See http://www.math.utah.edu/~yael/3210_public/exams/Integral.pdf) So in this case the Riemann integration technique can be used as well.