I'm proving the next proposition: Let $X$ be a random variable defined over a probability space $(\Omega,\mathcal{F},P)$ which takes values on $(E,\mathcal{E}),$ $\mathcal{G}$ a $\sigma-$algebra generated by a denumerable family $(Y_{n})_{n\geq1},$ which take values on $(Y,\mathcal{Y}),$ and let $f$ be a measurable real function, bounded or positive. A random variable $U$ $\mathcal{G}-$measurable is a version of the conditional expectation of $f(X),$ given $\mathcal{G},$ if $$E(f(X)\prod_{i=1}^{n}f_{k}(Y_{k}))=E(U\prod_{i=1}^{n}f_{k}(Y_{k})),$$ for every $n\geq 1$ and $f_{1}\ldots f_{n}$ functions $\mathcal{Y}-$measurable positive or bounded.
My idea is work with functional monotone class theorem. To do this, let $H$ be the set of all $U-\mathcal{G}$ measurables such that satisfie the equation of the proposition. I want to prove that $H$ is a linear vector space, but I have troubles proving the closedness because the sum of two elements in $H$ doesn't belong. The same with product by a element.
Any kind of help is thank in advance.