Horan's example (Horan 2020). Distinct kernels $V_ω$ are at distance at least $1$ in $𝒢X$, and $V_ω = V_{ω'}$ only for $ω' = ±ω$, so a measurable $W$ would make every set $S = -S$ measurable. The product $σ$-algebra is generated by countably many cylinders and so has at most $𝔠$ members, while the symmetric sets number $2^𝔠$.