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.