30/09/24
In general terms, measure theory is about measuring the size of sets. This could a geometric notion of size such as length, area, volume, or more general notion of magnitude: mass, charge, probability, etc. The goal is to develop a unified, abstract mathematical framework for dealing with all such measurements. Measure theory is the foundation of modern analysis, especially integration theory, Fourier analysis, and differential equations, as well as probability theory.
Towards the end of the 19th century it became clear that there are multiple problems with the established mathematical notions of magnitude (area, integral, probability, etc.) and measure theory was created by Lebesgue in response to this crisis. Here are some of these problems.
There was already a well-developed theory of integration developed by Riemann. However, in analysis we often want to take limits or infinite sums of functions and write things like
$\lim_{n\to\infty} \int f_n = \int \lim_{n\to\infty} f_n$
and
$\sum_{n=1}^\infty \int f = \int \sum_{n=1}^\infty f_n$
It turns out that the Riemann integral is not very well behaved under such operations. These formulae are true for the Riemann integral but only under very strong assumptions on the convergence of $f_n$ (recall uniform vs pointwise convergence from analysis). Moreover, the right-hand side might not even makes sense as the pointwise limit of Riemann integrable functions is not necessarily pointwise.
Example (Dirichlet function). A famous counterexample is the Dirichlet function $f \colon [0,1] \to \R$ is defined by $f(x) = 1$ if $x$ is rational and $f(x)=0$ is $x$ is irrational. This function is not Riemann integrable: if you recall the definition of the Riemann integral, it is the limit of the upper integrals and lower integrals if they agree. However, here for any division of $[0,1]$ into subintervals the upper integral is $1$, because any interval contains a rational number, but the lower integral is $0$, because any interval contains an irrational number.
The Dirichlet function is a pointwise limit of Riemann integrable functions. The rationals are countable meaning that we can order them in a sequence $\mathbb{Q} = \{ a_1, a_2, a_3, \ldots \}$. Define $f_n$ by$f(x) = 1$ for $x = a_1, a_2, \ldots, a_n$ and zero otherwise. Each $f_n$ is Riemann integrable with
$\int_0^1 f_n = 0$
but $f = \lim_{n\to\infty} f_n$ is not integrable. In some sense this tells us that $f$ wants to be integrable with
$\int_0^1 f = \lim_{n\to\infty} \int_0^1 f_n = 0$
but this cannot be done in Riemann integration theory.
You might object that such functions never appear in nature, it is just an artificial example. So we should ask: what are the assumptions on $f_n$ and $f$ such that we can take limits of integrals? How can we rule out such pathological examples? This is exactly the question that measure theory answers.
Example (Dirac delta). Another example, which is ubiquitous in physics and was discovered by a physicist, is the Dirac delta. Consider the Gaussian function with variance $\sigma$:
$f_\sigma (x) = \frac{1}{\sigma\sqrt{2\pi}} \exp(-\frac{x^2}{2\sigma^2})$
What happens when $\sigma \to 0$? The integral of these probability distributions is always $1$ but the Gaussians become more and more steep. We would want to say that these probability distributions converge to a probability distribution for which $0$ is true with probability $1$. (In quantum mechanics, this is like the limit from quantum physics to classical physics where all indeterminism disappears.) But of course these functions don’t converge. Measure theory gives us a way of taking such limits.
Examples and questions like these are fundamental in the study of differential equations, dynamical systems, Fourier analysis, and other branches of analysis.
We know how to define length, area, volume for simple geometric shapes such as polyhedra. Riemann integration allows us to extend these definitions to curved shapes such as surfaces. But this suffers from the same problems that Riemann integration theory itself. We would want to be able to take limits of shapes. As a result, we can obtain a shape that is not as regular as the original shapes, and our notion of length\area\volume based on Riemann theory breaks.
Example (Plateau’s problem). A classical problem in geometry is the Plateau problem: given a closed curve in $\R^3$ prove that there exists a unique surface of minimal area whose boundary is that curve. How to prove such a theorem? A natural approach would be by taking limits: take any surface $S_0$ whose boundary is a curve. If it has a minimal area among such surfaces, we are done. If not, then there is $S_1$ whose area is strictly smaller. Continue in this way to produce a sequence $S_1, S_2, \ldots$ of surfaces with smaller and smaller area. We would want to show that there exists a limit $S_\infty = \lim_{n\to\infty} S_n$ which is a solution to the Plateau problem. But how to take such limits? The limiting shape could be a priori very irregular, in the same way the limit of continuous functions can be discontinuous, so we might not even have the notion of area for such a shape. Measure theory allows us to take such limits and, in fact, prove that the Plateau problem has a unique solution. This was a major development in the history of geometry and calculus of variations.