I'm studying probability following Shiryaev's book and I came up with the following question, which I don't seem to find a proper example anywhere.
Because of the formulation of the conditional expectation of a random variable $\xi$ given a $\sigma$-algebra $\mathcal{G}$, under the conditions of the Radon-Nikodym theorem, I'm ensured that $E(\xi|\mathcal{G})$ always exists. Shiryaev shows (page 255) in fact why.
Radon-Nikodym theorem however is a theorem of existence: it ensures that there is a $\mathcal{G}$-measurable random variable $E(\xi|\mathcal{G})$ so that:
$$ E(\xi|\mathcal{G}) = \dfrac{d\mathbb{Q}}{d\mathbb{P}} (\omega), $$ where $\mathbb{Q}$ is a charge given by $$ \mathbb{Q}(A) = \int_A \xi d\mathbb{P}$$ and this is true up to sets of $\mathbb{P}$-measure zero, whenever I know that $\int \xi d\mathbb{P}$ is defined and $\mathbb{Q}$ is absolutely continuous w.r.t. $\mathbb{P}$.
However when I think and/or trying to find examples of $E(\xi|\mathcal{G})$, the construction is always explicit, meaning that I always find a way to obtain the random variable $E(\xi|\mathcal{G})$ either by finding a random variable $\eta$ that generates $\mathcal{G}$ or by the Radon-Nikodym theorem: evaluating $\mathbb{Q}(A)$ for $A \in \mathcal{G}$ and then taking the derivative $\dfrac{d\mathbb{Q}}{d\mathbb{P}}$, which I always seem to find an explicit form.
My question is: is there a case where I cannot find an explicit form for $E(\xi|\mathcal{G})$ and just know that it exists? If so, which example is that? Thanks!