I have problems determining what $\mathbb{E}(X| \mathcal{G})$ should be for $$X(\omega) = \sqrt{ \omega}, \ \ \ \mathcal{G}= \sigma \left( [0, 1/2 ), \ (1/3, 1] \right) $$
The random variable $X$ is defined on $\Omega = [0,1]$ with sigma algebra being all borel sets on $[0,1]$ that is $\mathcal{B}([0,1])$.
I've already done similar exercises, but there I had to find $\mathbb{E}(X|Y)$. There I wrote down what $\sigma (Y)$ was and tried to express variable $X$ using $Y$ and integrating $X$ over the generators of $\sigma (Y)$.
Here my problem is that I do not know how to go about finding or guessing $\mathbb{E}(X| \mathcal{G})$.
I've integrated $X$ over $\left[0, \frac{1}{2} \right)$ and $\left( \frac{1}{3}, 1 \right]$ but that didn't give me much.
I know that the conditional expectation should be measurable with respect to $\mathcal{G}$. And $\mathcal{G}$ consists of $\left[0, \frac{1}{2} \right), \ \left( \frac{1}{3}, 1 \right], \ \emptyset, \ \left[0,1 \right], \ \left(\frac{1}{3}, \ \frac{1}{2} \right), \ \left[\frac{1}{2}, 1 \right], \ \left[0, \ \frac{1}{3} \right], \ \left[0, \ \frac{1}{3} \right] \cup \left[\frac{1}{2}, 1 \right]$.
Could you tell me what I should do to find the conditional expectation?