= Conditional characteristic function
{title2=$\mathbb E[e^{i\theta Y}\mid\mathcal G]$}
For a real <random variable> $Y$ and a sigma-algebra $\mathcal G$, the conditional characteristic function is $\mathbb E[e^{i\theta Y}\mid\mathcal G]$. If it equals the nonrandom <characteristic function> of a probability law for every $\theta$, then $Y$ has that law and is independent of $\mathcal G$. To verify the independence assertion, multiply the identity by any bounded $\mathcal G$-measurable test variable and use the <uniqueness theorem for characteristic functions> on the resulting finite measures. Equality for rational $\theta$ extends by continuity, so simultaneous null sets can be arranged when needed.
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