= Characteristic-function derivative under an absolute first moment
{title2=$\phi\prime(0)=i\mathbb E X$}
If $\mathbb E|X|<\infty$, the <characteristic function> $\phi(u)=\mathbb E e^{iuX}$ satisfies $\phi'(0)=i\mathbb E X$. The bound $|(e^{iuX}-1)/u|\leq|X|$ and the <dominated convergence theorem> justify differentiation at zero. For mean zero, $\psi(u)=|1-\phi(u)|/|u|$ extends continuously by $\psi(0)=0$, and $|1-\phi(u)^n|\leq n|u|\psi(u)$ by a finite <geometric series>. This supplies a direct characteristic-function proof of the <weak law of large numbers>.
// Target: martingale.bigb
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