Let $\left(\Omega,2^\Omega,P\right)$ be a finite probability space. Let $P(\omega)>0 $ for all $\omega \in \Omega$. Let $X$ be a random vector, i.e. a measurable map $\Omega\to\mathbb R^n.$ Show that if $\mathcal B \subset \mathcal A$ is a σ-algebra containing $\sigma(X)$ and if $X'$ is another $\mathcal A$-measurable random vector of dimension $n'\in\mathbb N$, then, for any Borel function$ f : \mathbb R^{n+n'} \to\mathbb R$, we have $$E[(f(X,X')\mid \mathcal B](\omega)=E[f(X(\omega),X')\mid \mathcal B](\omega)$$ for any $\omega\in\Omega$.
I do not see how to handle this problem. Any help is welcome!
First note that every $\mathbb{R}$-valued random variable on a finite probability space is integrable.
Denote the class of Borel functions on $\mathbb{R}^{n+n'}$ that satisfy the equality by $\mathcal{F}$.
Pick a Borel subset $A_1\in \mathcal{B}(\mathbb{R}^n)$ and a Borel subset $A_2\in \mathcal{B}(\mathbb{R}^{n'})$. Then
$$\mathbb{E}\big[\chi_{A_1}(X)\cdot \chi_{A_2}(X')\,\big|\,\mathcal{B}\big] = \chi_{A_1}(X)\cdot\mathbb{E}\big[\chi_{A_2}(X')\,\big|\,\mathcal{B}\big]$$
because $\chi_{A_1}(X)$ is an integrable random variable measurable with respect to $\mathcal{B}$ (c.f. this property is called "pulling known factors" in this wikipedia article). Hence
$$\mathbb{E}\big[\chi_{A_1}(X)\cdot \chi_{A_2}(X')\,\big|\,\mathcal{B}\big](\omega) = \chi_{A_1}(X)(\omega)\cdot\mathbb{E}\big[\chi_{A_2}(X')\,\big|\,\mathcal{B}\big](\omega) = $$ $$=\bigg(\chi_{A_1}(X)(\omega)\cdot \mathbb{E}\big[\chi_{A_2}(X')\,\big|\,\mathcal{B}\big]\bigg)(\omega) = \bigg(\mathbb{E}\big[\chi_{A_1}(X)(\omega)\cdot \chi_{A_2}(X')\,\big|\,\mathcal{B}\big]\bigg)(\omega)$$
Thus $f(x,x') = \chi_{A_1}(x)\cdot \chi_{A_2}(x')$ in in $\mathcal{F}$. By the usual properties of conditional expectation the equality holds for all Borel functions. Indeed, $\mathcal{F}$ contains all nonnegative linear combinations of functions of the form $$(x,x')\mapsto \chi_{A_1}(x)\cdot \chi_{A_2}(x')$$ By monotone convergence for conditional expectations we deduce that $\mathcal{F}$ is closed under monotone convergence of its nonnegative memebers. Finally the equality holds for differences of functions in $\mathcal{F}$. Now the fact that $\mathcal{B}(\mathbb{R}^{n+n'})= \mathcal{B}(\mathbb{R}^n)\otimes \mathcal{B}(\mathbb{R}^{n'})$ and the properties described above imply that $\mathcal{F}$ contains all Borel measurable functions on $\mathbb{R}^{n+n'}$.