Here we have a big list of great applications of the Baire category theorem.
I recently read the Birkhoff ergodic theorem and I think perhaps this theorem is on par with Baire's theorem in terms of its applications to diverse topics.
The theorem states that (See Theorem 1.14 in Peter Walter's An Introduction to Ergodic Theory).
Let $(X, \mathcal F, \mu)$ be a $\sigma$-finite measure space and $T:X\to X$ be a measure preserving transformation. Let $f\in L^1(X, \mu)$. Then $(1/N)\sum_{n=0}^{N-1}f(T^nx)$ converges almost everywhere to an $L^1(X, \mu)$-function $f^*$. Further, we have $f^*\circ T=f^*$ and if $X$ is a finite measure space than $\int_Xf^*\ d\mu=\int_Xf\ d\mu$.
These are some applications of the Birkhoff ergodic theorem that I know of:
- Here is a proof of the law of large numbers using the Ergodic theorem
- If $P$ is the transition matrix of a finite state space Markov chain having a strictly positive stationary distribution, then the limit $$ Q:=\lim_{N\to \infty} \frac{1}{N} \sum_{n=0}^{N-1} P^n $$ exists. The matrix $Q$ is a stochastic matrix and satisfies $QP=PQ=Q$, and $Q^2=Q$. This has applications to Markov chains. A proof can be found in Peter Walter's An Introduction to Ergodic Theory (Lemma 1.18)
- Almost all numbers in $[0, 1)$ are normal in base 2, that is, for almost all $x$ in $[0, 1)$ the frequency of $1$'s in the binary expansion of $x$ is $1/2$. (For a proof see Theorem 1.15 in Peter Walter's An Introduction to Ergodic Theory).
What is your favourite application of the ergodic theorem?
For almost every $x \in [0,1]$, the elements of the continued fraction expansion of $x$ are unbounded.
This can be seen by studying $\mathbb{R}/\mathbb{Z}$ with $\frac{1}{\log 2}\frac{1}{1+x}dx$ as the measure. Let $T: \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ be $Tx = \{\frac{1}{x}\}$, the fractional part of $\frac{1}{x}$ (with $T0 := 0$), and $f(x) = \lfloor \frac{1}{x}\rfloor$. The ergodic theorem says that, if $x = [x_1,x_2,\dots]$, then $$\frac{x_1+x_2+\dots+x_N}{N} = \frac{1}{N}\sum_{n=0}^{N-1} f(T^n x) \to \int_0^1 f(y)\frac{1}{\log 2}\frac{dy}{1+y} = +\infty.$$ This tells us a bit more than unboundedness, but when I first heard that the result is proven by ergodic theory, I was shocked how this analytic machinery could prove this very number theoretic statement [I didn't know much analytic number theory at the time either :P]. Of course "almost every" appears in the statement of the result, but I definitely don't view that as a big restriction. This result is what made me want to study ergodic theory.