Let $\varphi$ be the ch. f of $\mu$. After proving the inequality $$\mu \left\{ x : |x|>\frac{2}{u}\right\}\leq \frac{1}{u}\int_{-u}^u(1-\varphi(t))\,dt$$ the text im reading states "this shows that the smoothness of the characteristic function at $0$ is related to the decay of the measure at $\infty$." I can see that the left hand side refers to the decay of $\mu$ at infinity, but I don't understand exactly what is the relationship between the right hand side and the "smoothness" of $\varphi$.
I guess my question might be: how is smoothness measured exactly and how does that measure of smoothness relate to the right hand side of the inequality above?