 Measurable-factor rule for conditional expectation

Let $Y$ be an integrable random variable, let $H$ be a finite $\mathcal{G}$-measurable random variable, and suppose $HY$ is an integrable random variable. Then a known factor can be taken outside the conditional expectation:For a bounded $H$, first verify the defining integral identity when $H$ is the indicator function of a measurable set in $\mathcal{G}$, extend by linearity to simple functions, and then use bounded approximation and the dominated convergence theorem. For nonnegative $H,Y$, truncating $H$ and using the conditional monotone convergence theorem proves the identity with extended nonnegative conditional expectations. In particular,For general $H,Y$ under the stated integrability assumptions, the conditional Jensen inequality bounds $|H\mathbb{E}[Y\mid \mathcal{G}]|$ by that integrable random variable. Apply the bounded-factor result to truncations of $H$, then use the dominated convergence theorem on the products and the L1 contraction of conditional expectation. This proves the identity without assuming $H$ itself is an integrable random variable.

The product-integrability condition is essential to the ordinary finite conditional expectation statement. An unbounded factor cannot be moved through an expected value simply because its partner is an integrable random variable.^[ref]