TITLE: Extending the product measure on $2^\omega$
QUESTION [8 upvotes]: Consider the standard completed product measure $P$ on $\Omega=\{0,1\}^\omega$ corresponding to an i.i.d. sequence of fair coin-flips. 
Given $n\in\omega$, let $\rho_n$ be the bijection of $\Omega$ to itself given by flipping the value of the $n$th coordinate.
Say that a subset $A$ of $\Omega$ has property $H$ iff there is an $n$ such that $\rho_n(A)\cap A=\varnothing$ and $\rho_n(A)\cup A=\Omega$. 
Assume AC.
Question: What can we say about the existence of (countably-additive) extensions of $P$ that assign measure $1/2$ to all subsets with property $H$? Are there any? If so, are there any with nice invariance properties, like invariance under $\rho_n$ and/or under bijections of $\{0,1\}^\omega$ induced by permutations of $\omega$?
Remarks: Intuitively, the probability of $A$, if $A$ has property $H$, should be $1/2$. This is true with respect to $P$ if $A$ is $P$-measurable. But given AC, there are $P$-nonmeasurable $A$ having $H$. Say that $\alpha\sim\beta$ iff one can get $\beta$ from $\alpha$ by applying an even number of the $\rho_n$. Given a $\sim$-equivalence class $U$, let $U' = \rho_1 U$. Now take a set $B$ of equivalence classes that contains exactly one member of $\{ U,U' \}$ for every $U$, and let $A$ be the union of the members of $B$.  Then $A$ has $H$ with respect to every index $n$, and hence is non-measurable by Lévy's zero–one law.

REPLY [2 votes]: There is no such measure. Suppose toward contradiction $\mu$ is such a measure.
Partition $\Omega$ by the mod finite equivalence relation $\sim_{\text{fin}},$ let $X \subset \Omega$ be a choice of representatives and $\pi: \Omega \rightarrow X$ send each sequence to its representative.
Let $X_n = \{x \in \Omega: x \triangle \pi(x) \subset [n]\}.$ Each $X_n$ can be extended to some $H_n$ satisfying $\Omega \setminus H_n = \rho_{n+1}(H_n),$ and therefore having property $H.$ For each $x \in X,$ there are cofinitely many $n$ such that $x \in X_n \subset H_n.$ Thus, $\Omega = \bigcup_{n \in \mathbb{N}} \bigcap_{m \ge n} H_m.$ This is an increasing union of sets of $\mu$-measure bounded by $\frac{1}{2},$ contradiction.