= Weak-star limit of periodic rescalings
{title2=$\theta(j\cdot)\overset*\rightharpoonup\int_0^1\theta$}
A <bounded> measurable one-periodic <function> rescaled to frequency $j$ converges <weak-star topology>[weak-star] in $L^\infty(\mathbb R)$ to its period average. Subtract the average and take a <bounded> periodic primitive $F$. Its <integral> on $[a,b]$ after rescaling is $[F(jb)-F(ja)]/j$, which tends to zero. Sequential Banach-Alaoglu with separable predual $L^1$ then identifies every cluster point by <interval> <integrals>; uniqueness of all cluster points gives full <weak-star topology>[weak-star convergence]. Localizing to <bounded> <intervals> avoids falsely assuming that a <bounded> cluster <function> is globally integrable.
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