Translation Invariant Random Variable (Percolation)

Question

I'm currently studying the uniqueness of the infinite cluster in supercritical percolation. In the proof the rv $N$ counts the number of infinite clusters on the infinite 2-D square lattice. The claim is that $N$ is translation invariant (obvious) and that this implies it is constant almost surely (in the extended reals). I'm having trouble verifying the implication that it must be constant. Is this a sort of Hewitt Savage 0-1 law? Except that for Hewitt Savage I need invariance under finite permutations as well. I feel like the proof should be obvious but I'm not seeing it. Can anyone offer a hint in the right direction? Thanks!

Answer 1

You are right, it is a sort of Hewitt-Savage zero-one law, but not exactly.

Following Grimmett, take $\Omega=\{0,1\}^{\mathbb{E}^d}$ with the natural family of translations inherited from the translations of the lattice $\mathbb{L}^d$. Since we are interested in bond percolation, $P_p$ is the product of Bernoulli($p$) measures on $\Omega$.

You need the fact that the $\sigma$-field $\cal T$ of translation invariant events is $P_p$ trivial. This is proved with exactly the same argument used to proved the Hewitt-Savage zero-one law, except the map on the index set is a translation instead of a finite permutation.