Weak-star limit of periodic rescalings ($\theta (j\cdot )\stackrel{\ast }{\rightharpoonup }{\int }_0^1\theta$)

A bounded measurable one-periodic function rescaled to frequency $j$ converges 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 convergence. Localizing to bounded intervals avoids falsely assuming that a bounded cluster function is globally integrable.