I cite this textbook, chapter $6$.
Some definitions:
A measure preserving system is a probability space $(X,\Sigma,\mu)$ equipped with a measurable "dynamic" $\varphi:X\to X$ such that $\mu\equiv\mu\circ\varphi^{-1}$ everywhere, which intuitively represents the evolution over time.
The "measure algebra" $\Sigma(X)$ is the quotient set $\Sigma_{/\sim}$ where $A\sim B\iff\mu(A\setminus B)=0=\mu(B\setminus A)$. Here, we define the "induced map" $\varphi^\ast:\Sigma(X)\to\Sigma(X)$ by $[A]\mapsto[\varphi^{-1}A]$, which is well-defined as $\varphi$ preserves null sets. All the following (in)equalities will be in this a.e. equivalence sense and sets should be interpreted as elements of $\Sigma(X)$ unless otherwise stated.
We say that a set $A\in\Sigma(X)$ is recurrent if: $$A\subseteq\bigcup_{n\ge1}\varphi^{\ast n}A$$And infinitely recurrent if: $$A\subseteq\bigcap_{k\ge1}\bigcup_{n\ge k}\varphi^{\ast n}A$$
For null sets, their $\Sigma(X)$ equivalence class is simply that of all other null sets so recurrence is not meaningful, but for sets of positive measure (infinite) recurrence means that almost all points will (infinitely often) return to the set over time.
Poincare's theorem is that every element of $\Sigma(X)$ is infinitely recurrent. I wish to emphasise recurrence; studying the definition, we find that recurrence here defined means that almost every point in $A$ is mapped over time back into $A$, but it says nothing about $A$ occurring in the first place. That is, given that we observe event $A$ in the state space, we can be almost sure we will observe the same event again, infinitely often.
The text calmly discusses these results as statements also of occurrence which is not sitting well with me.
The chapter begins with a quote from Nietzsche about recurrence: in this spirit they conclude the chapter with a whimsical exercise:
The book “Also sprach Zarathustra” by F. Nietzsche consists of roughly $680,000$ characters, including blanks. Suppose that we are typing randomly on a typewriter having $90$ symbols. Show that we will almost surely type Nietzsche’s book (just as this book you are holding in your hand) infinitely often. Show further that if we had been typing since eternity, we almost surely already would have typed the book infinitely often. (This proves correct one of Nietzsche’s most mysterious theories, see the quote at the beginning of this chapter.)
The expected approach is to use the fact that typing can be modelled as a Bernoulli shift over the alphabet of $90$ symbols and the "event" of Nietzsche's book is the cylinder-set of sequences beginning with the first $\approx680,000$ characters of the book and all terms thereafter being arbitrary symbols. The Bernoulli shift system is measure-preserving (even ergodic) so we can say that all events, namely Nietzsche's book, will recur ad infinitum, where "typing" is modelled by shifting one character over in the sequence. However, this construction, it seems to me, is conditional on us starting with an element of the book-event (any sequence starting with the book's text). Were we to start with, to invoke the other apocryphal random typewriter experiment (infinite monkey theorem? :) ), Shakespeare's Hamlet, I think we can no longer be almost sure of also typing Thus Spake Zarathrustra.
Am I right? The text seems to imply here (and elsewhere, but this was the only example I could find) that regardless of the initial state we would almost surely have typed the book and almost surely will continue to type the book, which seems dissonant with the actual definition of recurrence.