Conditional characteristic function ($\mathbb{E}[e^{i\theta Y}\mid \mathcal{G}]$)

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.