Since $𝓛_ω$ has rank one and $𝓛_ω e₀ = ±e₀$, $‖𝓛^{(n)}_ω x‖ = |φ_ω(x)|$ for $n ≥ 1$, so $λ₁ = 0$, $λ₂ = -∞$, and the slow space is $ker φ_ω$. By density of $ℤ√2$ in the circle the sign patterns $(sᵢ(ω))ᵢ$ separate points, and kernels of distinct functionals $φ_ω ≠ ±φ_{ω'}$ are at distance at least $1$ in $𝒢X$. A measurable choice of slow spaces would therefore make every subset of $[0, 1/2)$ measurable, but the Borel $σ$-algebra of the circle has only $𝔠$ members.