Consider bond percolation on $\mathbb{Z}^d$. How can we prove that the set of configurations $A = \{ \text{there exist an infinite open cluster} \}$ is an event, i.e. that it belongs to the cyllinder sigma-field ?
If we assume that every edge is open/closed with probability $p$ independently of all other edges, then Kolmogorov's $0$-$1$ law states that the event $A$ either has probability $0$ or $1$ of happenning. Is there an expression involving countable unions/intersections that make its belonging to the tail sigma-field obvious ?
The complement of the event is that all clusters are finite. Thus $A^c$ is a countable union of sets of the form $B_x=\{\text{the cluster around } x \text{ is finite}\}$. Thus it suffices to show each of those is measurable. There are only countably many finite clusters around $x$, so $B_x$ is a union of $C_{F} = \{\text{the cluster around } x \text{ is the cluster } F\}$ where $F$ ranges over all finite clusters around $x$. Thus it suffices to show each of those is measurable. The complement of $C_F$ is the union of the events that $y$ is adjacent to $z$, where $y\in F, z \notin F$, union the events that the cluster is too small. In order to capture the "too small" event precisely, we can look at $2^{|F|}$ cylinder sets specifying the edges between elements of $F$, and pick out those which do not in themselves imply that the component around $x$ is at least as large as $|F|$.