I am struggling to understand how to compute the Bochner integral that furnishes us with the conditional expectation of a Banach-valued random variable.
In particular, I consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a Banach space $X$ with $X=C[0,1]$, i.e. the set of continuous functions on the closed unit interval. Now, I am interested in a measurable function (random variable) $Y:\Omega\to C[0,1]$.
I am wondering about the conditional expectation given as $$\int_AE[Y|\mathcal{G}]d\mathbb{P}=\int_AYd\mathbb{P}\quad\forall\quad A\in\mathcal{G}$$ where the integrals are Bochner integrals.
To make my question more tangible, suppose that $Y(\omega)$ is a generalized Brownian bridge from, say, $a$ to $b$. Hence, we'd have that $$Y_t=(1-t)a+tb+Z_t$$ where $Z_t$ is a standard Brownian bridge. Suppose we observe the realisations of $Y_t$ for all $t\leq \tau$ with $\tau<1$ and let $Y^{\tau}$ denote the history of all observations of $Y_t$ up to $t=\tau$.
My intuition would suggest that $E[Y|Y^\tau]:[0,1]\to R$ is equal to $Y_t$ for all $t\leq\tau$ and a straight line from $Y_\tau$ to $b$ for all $t\geq \tau$.
Unfortunately, I am entirely at a loss when it comes to actually showing that this is true.
Many thanks and best wishes.