= Characteristic-function tightness bound
{title2=$\mu(|x|\geq2/h)\leq h^{-1}\int_{-h}^h(1-\operatorname{Re}\varphi(t))\,dt$}
For a <probability measure> on the real line with <characteristic function> $\varphi$, averaging $1-\cos(tx)$ over $[-h,h]$ gives $1-\sin(hx)/(hx)$. This is at least $1/2$ when $|x|\geq2/h$. Integrating gives the displayed bound. Consequently pointwise convergence of <characteristic functions> to a function continuous at zero implies <tightness of probability measures>, by the <dominated convergence theorem> on $[-h,h]$ and a separate tail bound for the finitely many initial measures.
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