# Conditional expectation of random vectors

$$\newcommand{\E}{\mathsf{E}}$$ $$\newcommand{\P}{\mathsf{P}}$$ The following additional question was asked in a comment by user Oleg:

Suppose that $$(\Omega,\mathcal F,\P)$$ is a probability space, $$B$$ is a (not necessarily separable) Banach space, and $$F\colon\Omega\to B$$ is a strongly measurable random vector in $$B$$, with $$\E\|F\|<\infty$$. If now $$\mathcal G$$ is a sub-σ-algebra of $$\mathcal F$$, is then there a well-defined and strongly measurable conditional expectation $$\E(F|\mathcal G)$$?

The strong measurability of a random vector $$F$$ means that there is a sequence of finitely-valued random vectors $$F_n$$ in $$B$$ such that $$\|F_n(\omega)-F(\omega)\|\to0$$ for $$\P$$-almost all $$\omega\in\Omega$$.

I thought it makes sense to post this question separately, which is what is being done here.

$$\newcommand{\E}{\mathsf{E}}$$ $$\newcommand{\P}{\mathsf{P}}$$ The affirmative answer to this question is provided by Scalora, Theorem 2.1, page 354, which can be stated as follows, using the setting and notation in the question:
If $$F\colon\Omega\to B$$ is a Bochner-integrable random vector, then there is a $$\P$$-almost surely unique Bochner-integrable strongly $$\mathcal G$$-measurable random vector $$\E(F|\mathcal G)=:G$$ in $$B$$ such that $$\E G1_A=\E F1_A$$ for all $$A\in \mathcal G$$.
Note that in Scalora's paper the strong measurability is part of the definition of the Bochner integrability. Note also that then, by Bochner's theorem, $$F$$ is Bochner-integrabile iff $$F$$ is strongly measurable and the condition $$\E\|F\|<\infty$$ holds.