Characteristic-function tail bound ($\mu (|y|\ge \lambda )\le \frac{\lambda }{1-\sin 1}{\int }_0^{1/\lambda }(1-\mathrm{Re}\varphi (u))du$)

Average $1-\cos (uy)$ over $0\le u\le 1/\lambda$. Multiplication by $\lambda$ gives $1-\sin (y/\lambda )/(y/\lambda )$, with value zero at $y=0$. This is at least $1-\sin 1$ when $|y|\ge \lambda$. Tonelli's theorem then gives the displayed bound for a probability measure with characteristic function $\varphi$. The bound converts control of the characteristic function near zero into tightness of probability measures and is useful in proving the Lévy continuity theorem.