Is measure theory only for integrals?

Question

I am trying to self-study probabilistic measure theory after completing my undergrad degree, and I am curious if there are more interesting applications of measure theory aside from Lebesgue integration ? It seems like (correct me if I am wrong here) measure on its own is a rich field before delving in integration of measurable functions.

Answer 1

I would propose that the construction of measures itself can be a highly non-trivial task in measure theory, yet it is the very foundation of it.

Think of what a measure is: It is a non-negative set-function $\mu$ from some $\sigma$-algebra $\mathcal F$. This means, to define a measure we need to assign to every set $A\in\mathcal F$ a value $\mu(A)\in[0,\infty]$. But often times the $\sigma$-algebra $\mathcal F$ will be very large and involve very complicated sets (for example Cantor set in the real line). Are we really going to find some nice "formula" to define $\mu(A)$ for every possible $A\in\mathcal F$? Probably not...

This is why extension Theorems like Caratheodory's Extension Theorem are so vital: They allow us to uniquely define measures on a "complicated" $\sigma$-algebra $\mathcal F$ by only defining it on some "simpler" generating set $\mathcal C\subset \mathcal F$. This would be one example of an important piece of measure theory that is "independent" of integration.

To give another interesting example from Probability Theory, where we are not so much concerned about integration with respect to a probability measure, but rather constructing a probability measure in the first place: Consider the theory of Stochastic Processes. Often times we want the paths of such a process to be continuous (or at least cádlág), which means we are trying to construct some probability measure on the set of continuous functions $C$ (or the set of cádlág functions $D$). But how would one even try to do this? Note that now our $\sigma$-algebra consists of "function sets", i.e. sets $A\subseteq C$.

The most intuitive way to understand a stochastic process is to not look at its law over $C$, but rather at the laws of the finite-dimensional projections of it. This means, one may try to construct a stochastic process $(X_t)_{t\geq 0}\in C$ by considering the finite-dimensional projections $(X_{t_1},\dots,X_{t_k})$ for various $t_1<\dots<t_k\in[0,\infty)$ and hope to "extend" the collection of their distributions on $\mathbb R^k$ to a distribution on $C$. Here, extension Theorems like Kolmogorov's Extension Theorem together with Kolmogorov's Continuity Theorem provide a way to do this.

Again, in the theory of stochastic processes we may not even be interested in integrating against a measure $\mu$ on $C$ (though there are certainly areas of Stochastic Analysis where this is done) but rather constructing such measures and analyzing them in some way, e.g. studying Weak Convergence on $C$. While weak convergence can be understood via integrationg w.r.t. bounded continuous function (à la Portmanteau), this approach is completely intractable for stochastic processes, hence one looks for other criterions of weak convergence, see e.g. here for an overview for continuous processes and here for an overview for cádlág processes.