Two questions about conditional expectations

Question

1. When $Y$ is $\mathcal{F}$-measurable, what are the minimal conditions for $E[XY|\mathcal{F}]=YE[X|\mathcal{F}]$ to hold? Is it enough to know that both $X$ and $XY$ are $L^1$ or do we need some condition on $Y$ as well, besides measurability (e.g. $Y\in L^1$)?
2. Building on the previous question, when considering the transform $(Y_n)_{n\in\mathbb{N}}$ of a martingale $(X_n)_{n\in\mathbb{N_0}}$ by a predictable $(H_n)_{n\in\mathbb{N}}$, i.e. $Y_n=X_0+\sum_{i=1}^{n}H_i(X_i-X_{i-1})$, why do we need boundedness assumptions on $H_n$ to prove $(Y_n)_{n\in\mathbb{N}}$ is a martingale (e.g. the $H_n$ are uniformly bounded or at the very least individually bounded)? In the proof the boundedness assumption seems to be used only to pull $H_n$ out of the conditional expectations, but, assuming that for this to work we'd only need measurability of $H_n$, why assume boundedness (and hence $H_n\in L^1$)?

Answer 1

(1). The most general formulation of the property I am aware of states that over a $\sigma$-finite measure space $(X,\mathscr{A},\mu)$, if $\mathscr{G}\subseteq \mathscr{A}$ is a sub-$\sigma$-algebra then $E[gu|\mathscr{G}]=gE[u|\mathscr{G}]$ for all functions $g $ which are $\mathscr{G}$-measurable and all $u \in L^{\mathscr{G}}(\mathscr{A})$, which is the maximal set of $\mathscr{A}$-measurable functions for which the conditional expectation is defined (ref. Schilling, 2017, Th. 27.11.(vii)).

(2). Recall martingales need to be $\mathscr{F}_n$-adapted, integrable $\forall n$ and the martingale property must hold. If $X_n,H_n$ are adapted, then $Y_n$ as defined is adapted as it is a linear combination of $\mathscr{F}_n$-measurable functions. Boundedness of $H_n$ (i.e. $|H_n|\leq M_n<\infty,\,M_n \in \mathbb{R}$) is used to prove integrability: $$E[|Y_n|]\leq E[|X_0|]+\sum_{k\leq n}E[|H_k||X_k-X_{k-1}|]\leq E[|X_0|]+\sum_{k\leq n }M_k(E[|X_{k}|]+E[|X_{k-1}|])<\infty$$ for the martingale property, just use the predictability of $H_n$ and you obtain $E[Y_n-Y_{n-1}|\mathscr{F}_{n-1}]=H_nE[X_n-X_{n-1}|\mathscr{F}_{n-1}]=0$ which shows the martingale property.