Existence of a conditional expectation construction for metric spaces

Question

Following the construction of the conditional expectation for real valued random variables, I wondered if it can be 'generalized' to metric spaces as follows. Let $(\mathbb{X}, d)$ be a metric space, $X:\Omega \to \mathbb{X}$ a random variable and $\mathcal{G}$ some subsigmaalgebra in $\Omega$. Is there in general a $\mathcal{G}$-measurable $Z:\Omega \to \mathbb{X}$ such that $\mathbb{E}[d(X,Z)\mathbb{1}_{A}] = 0$ for all $A\in \mathcal{G}$ ? I am not sure whether Radon-Nikodým can be applied here in some way.

Answer 1

What you want is $d(X,z)1_A=0$ a.s (because of non-negativity) and, in particular, this must holds for $A=\Omega$. So this is possible only if $X$ is a.s. equal to some $Z$ which is $\mathcal G-$ measurable.

Answer 2

If $X$ is a random variable on an arbitrary probability space $(\Omega, \mathcal{A}, \mathbb{P})$ and taking its values in a standard Borel space $\mathbb{X}$, and $\mathcal{B} \subset \mathcal{A}$ is a $\sigma$-algebra, then there exists a regular conditional distribution $\mathbb{P}(X \in \cdot \mid \mathcal{B})$ of $X$ given $\mathcal{B}$. A Polish space is a standard Borel space. The existence of the regular conditional distribution does not hold in general if we only assume that $\mathbb{X}$ is a separable metric space.

Then for any suitable function $f \colon \mathbb{X} \to \mathbb{R}$, one has the conditional expectation of $f(X)$ given $\mathcal{B}$: $$ \mathbb{E}[f(X) \mid \mathcal{B}](\omega) = \int f(x) \mathrm{d}\nu_\omega(x) $$ where $\nu_\omega$ is the probability measure $\mathbb{P}(X \in \cdot \mid \mathcal{B})(\omega)$.