= Translation continuity in L1 on the circle
{title2=$\|f(\cdot+t)-f\|_1\to0$}
For $f\in L^1(\mathbb T,m)$ with $m$ normalized <Lebesgue measure>, translation on the <circle group> is continuous in the <L1 norm>: $\|f(\cdot+t)-f\|_1\to0$ as $t\to0$ modulo one. Approximate $f$ in the <L1 norm> by a continuous function, use its <uniform continuity>, and use translation invariance to preserve the approximation error. In particular, $m(B\mathbin\triangle(B-t))\to0$ for every measurable $B$.
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