Let $(\Omega, \mathcal{A},P)$ be a probability space, $\mathcal{F} \subseteq \mathcal{A}$ a $\sigma$-algebra and $X:\Omega \rightarrow \mathcal{X}$ a random variable which takes values in a countable set $\mathcal{X}=\{x_1,x_2,\dots\}$.
Show that there exists a mapping $P_{\mathcal{F}}:\mathcal{P}(\mathcal{X}) \times \Omega \rightarrow [0,1]$ so that
i) $P_{\mathcal{F}}(\cdot,\omega)$ is a probabilty measure on $\mathcal{P}(\mathcal{X})$ for every $\omega \in \Omega$.
ii) $P_{\mathcal{F}}(A,\cdot)$ is a conditional expectation of $\mathbb{1}_{\{X\in A\}}$ given $\mathcal{F}$ for every $A \in \mathcal{P}(\mathcal{X})$.
I am struggling so far with this exercise and am in need of some help. I was given the hint to show that $E[\mathbb{1}_{\{X=x_i\}} \vert \mathcal{F}],i \in \mathbb{N}$ is a probabilty sequence and then construct the probabilty measure with this sequence. But to be honest I don't know how to proceed.
Any help would be appreciated, thanks in advance.