# Conditional expectation on Function space

This question is from a notation in section 13.4 of the book

"Linear and Nonlinear Filtering for Scientists and Engineers" By N U Ahmed

In this section, the author is deriving the Zakai Equation. I have a question about the notation, and I will try to formulate my question so that you do not have to refer back to the book.

Given the space of continuous functions $C([0,T], \mathbb{R}^n)$ with the sup-norm, let $\mathcal{B}(C)$ denote the $\sigma$-algebra of Borel subsets of $C$. We have a random variable $$X: (\Omega,\mathcal{F},P) \rightarrow (C,\mathcal{B}(C))$$ such that $\omega \mapsto X(\cdot, \omega)$.

The R.V. will give us a push forward probability measure on the measurable space $(C,\mathcal{B}(C))$ by $$\mu_X (E)= P(\{\omega\in \Omega : Y(\cdot,\omega) \in E\})$$ for $E\in \mathcal{B}(C)$.

Now the book stated "Let $E^X$ denote the integrations on $C$ with respect to the measure $\mu_X$", then it used the following notation $$E^X[g(X(t)) | \mathcal{H}]$$ where $g$ is a bounded Borel measurable function on $\mathbb{R}^n$ and $\mathcal{H}$ is a sub $\sigma$-algebra of $\mathcal{F}$ in original probability space.

I was wondering is this a conditional expectation on the original probability space or the function space? Since the $t$ appears in the expression, I would guess it is a conditional expectation on the original probability space, but how does $E^X$ work here? Maybe this is an abusive notation that I am not aware of.

If you would like to look at the book, it is here. This is where they used such notation:

The $\mu_X$ I defined is the $\mu_1$ in the book, and my $E^X$ is its $E^1$. We can ignore $\mu_0$ and $E^0$ for now, I was not sure how the middle equality between $E$ and $E^1$ is defined in the picture.

• As you guessed, this is wrong since $\mu_X$ can only integrate functions defined on $C$ while $g(X(t))$ is a function defined on $\Omega$. On the other hand, the notation $E[g(X(t)) | \mathcal{H}]$ is correct, with $E$ the expectation with respect to $P$ the probability measure on $\Omega$, $g(X(t))$ some measurable function defined on $\Omega$, and $\mathcal H$ some sub-sigma-algebra of $\mathcal F$ on $\Omega$. (Where in the book is the passage you quote?) – Did Mar 28 '16 at 11:50