I have a problem with the following exercise.
Let $(\Omega, \mathcal{H}, \mathbb{P})$ be the probability space defined by $\Omega = (0,1)$, $\mathcal{H} = \mathcal{B}(0,1)$ and $\mathbb{P}$ is the Lebesgue measure. Consider the following random variables: $X(\omega)=\omega^{\,2}$, $Y(\omega)=\omega(1-\omega)$ for each $\omega \in \Omega$. I have to calculate $\mathbb{E}[X|Y]$ but I do not know how to proceed.
Every suggestion or help is appreciated.
For every $B\in\mathcal B$ we have the equality:$$\int^{0.5}_{-0.5}z1_B\left(0.25-z^2\right)dz=0\tag1$$
Substituting $\omega=0.5-z$ leads to:$$\int^1_0\left(\omega-0.5\right)1_B\left(\omega\left(1-\omega\right)\right)d\omega=0\tag2$$Of course we have: $$\omega-0.5=\omega^2-\left[0.5-\omega(1-\omega)\right]=X(\omega)-\left[0.5-Y(\omega)\right]\tag3$$ so $(2)$ is equivalent with:$$\int^1_0X\left(\omega\right)1_B\left(Y\left(\omega\right)\right)d\omega=\int^1_0\left[0.5-Y(\omega)\right]1_B(Y(\omega))d\omega\tag4$$
We can write this also as:$$\int_{\{Y\in B\}}X(\omega)d\omega=\int_{\{Y\in B\}}0.5-Y(\omega)d\omega\tag5$$
This being true for every $B\in\mathcal H$ states exactly that: $$\mathbb E(X\mid Y)=0.5-Y$$