I am sorry that this is a rather trivial question but would very much appreciate a short answer.
Let $(\Omega,P(\Omega),\mathbb P)$ be a probability space where $\Omega=\{1,2,3,4,5,6\}$ and $\mathbb P(1)=\mathbb P(2)=\frac{1}{16}$, $\mathbb P(3)= \mathbb P(4)= \frac{1}{4}$ and $\mathbb P(5)=\mathbb P(6)=\frac{3}{16}$. Let X and Y be two Random Variables on $\Omega$ defined by $X=2 \cdot\mathbb{1}_{\{1,2\}}+8 \cdot\mathbb{1}_{\{3,4,5,6\}}$ and $Y=4\cdot\mathbb{1}_{\{1,2,3\}}+6\cdot\mathbb{1}_{\{4,5,6\}}$.
Compute $\mathbb{E}[X|Y=4]$ and $\mathbb{E}[X|Y]$
I have no problems calculating $\mathbb{E}[X|Y=4]$ but am somewhat hazy about $\mathbb{E}[X|Y]$. I know it is supposed to be a random variable $g(Y)$ and that $\mathbb{E}[X|Y]$ is defined through $$ \mathbb{E}[X|Y]: \Omega\to\mathbb{R}$$ $$ \omega\to\mathbb{E}[X|Y=Y(\omega)] $$
So would {1,2,3} then be projected onto $\mathbb{E}[X|Y=4]$ and {4,5,6} onto $\mathbb{E}[X|Y=6]$?
Then my random variable $\mathbb{E}[X|Y]$ would be $$\mathbb{E}[X|Y]=g(Y)=\mathbb{E}[X|Y=4]\cdot\mathbb{1}_{\{1,2,3\}}+\mathbb{E}[X|Y=6]\cdot\mathbb{1}_{\{4,5,6\}}$$
???
No further questions. Answer is given in the question.