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A nonlinear mapping on $L^2(S^1)$ that commutes with all translation operators is necessarily measurable?
Let $H:= L^2(S^1)$, where $S^1$ is the circle, and $\tau_a : H \to H$ be the translation operator for each $a \in S^1$:
\begin{equation}
(\tau_a f)(x):= f(x+a)
\end{equation}
Then, it is clear that ...
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