Let $G = \mathbb Z_2^\omega$, with pointwise addition. Assume the Axiom of Choice. I am interested in finitely additive probability measures $\mu$ defined on all of $\mathcal PG$ that can be intuitively thought to represent an infinite sequence of independent fair coin tosses.
One condition I want is $G$-invariance: $\mu(A)=\mu(\alpha+A)$ for $\alpha\in G$ and $A\subseteq G$. Since $G$ is abelian, there is an invariant measure on $\mathcal PG$.
Question: Is there a finitely additive invariant $\mu$ also subject to the independence condition that $\mu(A\cap B)=\mu(A)\mu(B)$ whenever $A$ and $B$ "depend on different coordinates".
(If we let $\pi_J : G\to \mathbb Z_2^J$ be the projection for $J\subseteq \omega$ given by $\pi_J(\alpha)=\alpha|_J$, then $A$ and $B$ depend on different coordinates iff we can write $A=\pi_J^{-1}[A']$ and $B=\pi_K^{-1}[B']$ for some disjoint $J$ and $K$.)
Comment: I was asked if there are $G$-invariant $\mu$ that don't satisfy independence. Yes. Start by noting that there is no $G$-invariant $\mu$ that is also invariant under the permutations of $\omega$ (acting by composition--I will write this action on the right). In fact, for any $G$-invariant $\mu$ on $G$ and infinite $A\subseteq \omega$ there will be a permutation $\tau$ fixing $\omega-A$ and a subset $C$ of $G$ depending only on the coordinates in $A$ such that $\mu(C)\ne\mu(C\tau)$.
Now, fix any $G$-invariant $\mu$, and let $A$ and $B$ be infinite disjoint subsets of $\omega$. It's easy to now show that there is a $G$-invariant $\nu$ (formed by combining $\mu$ with two permutations) and subsets $C$ and $D$ of $G$ depending on the coordinates in $A$ and $B$ respectively such that $\nu(C)\ne \mu(C)$ and $\nu(D) \ne \mu(D)$. Suppose $\mu$ and $\nu$ satisfy the independence condition. Replacing $C$ and/or $D$ with its complement as needed, assume $\nu(C)>\mu(C)$ and $\nu(D)>\mu(D)$. Let $\rho=(1/2)(\mu+\nu)$. Then $\rho$ is $G$-invariant but $\rho(C\cap D) > \rho(C)\rho(D)$, so $\rho$ does not satisfy the independence condition.