I'm reading the construction of conditional expectation w.r.t. a sub $\sigma$-algebra from this lecture note. The strategy is by Radon-Nikodým theorem. I would like to mimic this strategy for more general spaces. Could you have a check if my construction is valid?
First, we need some definitions of vector measures and RN property from Diestel-Uhl's Vector Measures.
Theorem 1. Let $\Sigma$ be a $\sigma$-field, $F: \Sigma \rightarrow X$ be a countably additive vector measure and $\mu$ be a finite nonnegative real-valued measure on $\Sigma$. Then $F$ is $\mu$-continuous, i.e., $\lim _{\mu(E) \rightarrow 0} F(E)=0$ if and only if $F$ vanishes on sets of $\mu$-measure zero.
Definition 4. Let $F: \mathscr{F} \rightarrow X$ be a vector measure. The variation of $F$ is the extended nonnegative function $|F|$ whose value on a set $E \in \mathscr{F}$ is given by $$ |F|(E)=\sup _\pi \sum_{A \in \pi}\|F(A)\|, $$ where the supremum is taken over all partitions $\pi$ of $E$ into a finite number of pairwise disjoint members of $\mathscr{F}$. If $|F|(\Omega)<\infty$, then $F$ will be called a measure of bounded variation.
Definition 3. A Banach space $X$ has the Radon-Nikodým property with respect to $(\Omega, \Sigma, \mu)$ if for each $\mu$-continuous vector measure $G: \Sigma \rightarrow X$ of bounded variation there exists $g \in L_1(\Omega, \Sigma,\mu, X)$ such that $G(E)=\int_E g \mathrm d \mu$ for all $E \in \Sigma$. A Banach space $X$ has the Radon-Nikodým property if $X$ has the Radon-Nikodým property with respect to every finite measure space.
Now we are ready to do the construction!
Let $(\Omega, \Sigma, \mathbb{P})$ be a probability space and $(X, |\cdot|_X)$ a Banach space. Here we use the Bochner integral. Let $f:\Omega \to X$ be $\mu$-integrable and $\mathcal G$ a sub $\sigma$-algebra of $\Sigma$. We want to define the conditional expectation $\mathbb E(f|\mathcal G)$ as an element in $L_1(\Omega, \mathcal G, \mathbb P, X)$ that satisfies $$ \int_A f \mathrm d \mathbb P = \int_A \mathbb E(f|\mathcal G) \mathrm d \mathbb P \quad \forall A \in \mathcal G. $$
Theorem: If $X$ has the Radon-Nikodým property, then $\mathbb E(f|\mathcal G)$ is well-defined and unique.
Proof: We define an $X$-valued vector measure $\mu$ on $(\Omega, \mathcal G)$ by $$ \mu (A) := \int_A f \mathrm d \mathbb P \quad \forall A \in \mathcal G. $$
Let $\pi$ be a partition of $\Omega$ into a finite number of pairwise disjoint members of $\mathcal G$. Then $$ \begin{align} |\mu| (\Omega) &= \sup_{\pi} \sum_{B \in \pi} |\mu(B)| &&= \sup_{\pi} \sum_{B \in \pi} \left | \int_B f \mathrm d \mathbb P \right |_X \\ &\le \sup_\pi \sum_{B \in \pi} \int_B \left |f\right |_X \mathrm d \mathbb P &&= \int_\Omega \left |f\right |_X \mathrm d \mathbb P \\ &= \|f\|_{L_1(\Omega, \Sigma, \mathbb P, X)} < \infty. \end{align} $$
Then $\mu$ is has bounded variation. It's clear that $\mu$ is $\mathbb P$-continuous. Because $X$ has RN property, there is $g \in L_1(\Omega, \mathcal G, \mathbb P, X)$ such that $$ \mu(A) = \int_A g \mathrm d \mathbb P \quad \forall A \in \mathcal G. $$
The fact that $g$ is unique up to a $\mathbb P$-null set $N \in \mathcal G$ is proved here. This completes the proof.
