The question is:
Let $(\Omega,\mathcal{F}, P) = ([0,1], R_{[0,1]},m)$ where $R_{[0,1]}$ is the $\sigma$-field of Borel subsets of the (closed) unit interval $[0,1]$, and $m$ is Lebesgue measure.
Let $\mathcal{G}$ be the $\sigma$-field generated by the events $[0,1/3]$,$(1/3, 2/3]$, and $(2/3,1].$ Define the random variable $X$ on $\Omega$ by $X(w)=w^2$. Find $E(X|\mathcal{G})$.
My attempt is:
The $\sigma$-field $\mathcal{G} = \{\phi, [0,1/3] , (1/3,2/3] \cup (2/3,1]~,(1/3,2/3],~[0,1/3]\cup (2/3,1],~(2/3,1],~[0,1/3] \cup (1/3,2/3],~[0,1]\}$.
In the question, $X(w)=w^2$, and I need to find a random variable $Y$ such that $E(X|\mathcal{G})=Y$ if
i) $Y$ is $\mathcal{G}$-measurable. ii) for all $A \in \mathcal{G}$, then $\int_{A} X(w) dw = \int_{A} Y(w) dw $.
Then I calculate the integration over the each element of $\mathcal{G}$ and I got $Y(w)=w^2$.
Is this a correct solution of this problem?