Finitely additive measures on $\mathbb Z_2^\omega$ with invariance and independence constraints

Question

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.

Answer 1

It's been a while since I've worked with amenable groups and integrals against finitely additive measures, so I could be missing something, but it now seems to me that the question is easy. While apparently in general the direct product of amenable groups isn't amenable, the direct product of abelian groups is of course amenable. So we can use:

Theorem. If $G$ is the direct product of the groups $(G_i)_{i\in I}$ and $G$ is amenable, then there is a finitely additive invariant measure on $G$ that has independence for finite sequences of events depending respectively on pairwise disjoint sets of coordinates.

We need only prove the Theorem for finite sets $I$. For let $F$ be the set of finite partitions of $I$, with a partial order given by $K\le L$ iff $L$ is at least as fine as $K$. For any partition $K\in F$, we consider $G$ to be the product of the $G^A=\prod_{i\in A} G_i$ as $A$ ranges over $K$ (here we use the fact that $G^A$ is amenable being a quotient of an amenable group). The measure $\mu_K$ we get from the finite index-set case of the theorem applied to the product of the $G^A$ will satisfy the independence condition for finite sequences of events depending on pairwise disjoint sets of coordinates that are in the algebra generated by $K$.

Let $\mu$ be a limit point of $\mu_K$ along the net $(F,\le)$. Then $\mu$ is the requisite measure on $G$.

Now on to the finite index-set case. Suppose we have amenable groups $G_1,...,G_n$ with invariant measures $\mu_1,...,\mu_n$. There is a f.a. measure on $G=\prod_i G_i$ satisfying the independence condition (e.g., a measure concentrated at a single point).

Suppose we have a f.a. measure on $G$ that satisfies the independence condition and is invariant under the actions of $G_1,...,G_k$ with $G_i$ considered as acting on the $i$th factor of $G$. Let $\mu'(A)=\int_{G_{k+1}} \mu(gA)\, d\mu_i(g)$. This is invariant under the actions of $G_1,...,G_{k+1}$.

Moreover $\mu'$ satisfies the independence condition. Suppose $A$ and $B$ depend on different sets of coordinates, and without loss of generality suppose $B$ does not depend on the $(k+1)$st coordinate. Then $\mu'(A\cap B)=\int_{G_{k+1}} \mu(gA)\mu(B)\, d\mu_i(g)=\mu'(A)\mu(B)=\mu'(A)\mu'(B)$, since $gB=B$ for $g\in G_{k+1}$.

Iterating we get a measure that satisfies the independence condition and is invariant under all of $G_1,...,G_n$.

[I realized that my original proof in the finite case didn't work, and I modified it along the lines of a suggestion in Uri Bader's comment.]