The problem asks to find $\mathbb{E}(\xi|\eta)$. But, following the same procedure as in other cases, it is necessary to determine $\sigma(\eta)$. Brzezniak' solution compute for $\mathbb{E}(\xi|\{\eta=1\})$, $\mathbb{E}(\xi|\{\eta=2\})$ and $\mathbb{E}(\xi|\{\eta=0\})$. But, given the general notation $\mathbb{E}(\xi|\eta)$ we need to compute for all sets in $\sigma(\eta)$, $2^3=8$ sets. Which includes cases like $\mathbb{E}(\xi|\{\eta=1\}\cup\{\eta=0\})$, etc. What do you think?
No; the notation $E(\xi \mid \eta)$ implies that the value of $\eta$ itself is given, and from the definition of $\eta$, it is clear that $$\eta : [0,1] \to \{0, 1, 2\}.$$ Thus the intent is to compute the conditional expectation for each value of $\eta$.
Moreover, to see why your interpretation is problematic, what would it mean to compute $E(\xi \mid \varnothing)$?
If the intent were to compute the conditional expectation for $\sigma(\eta)$ then the notation would have specified as such; e.g., $E(\xi \mid A)$ where $A \in \sigma(\eta)$.
