$\Omega$ = [0,1] $\mathbb{B}$ is the Borel $\sigma$-algebra and the $\mathbb{P}$ is the Lebesgue measure. If f(x) = x^2/2 , g(x) = 2(x-1/2)^2. 0 $\mathcal{F}$ = $\sigma{(f)}$ and $\mathcal{G} = \sigma{(g)}$ then what is $E[g|\mathcal{F}] , E[f|\mathcal{G}] $ ?
As to generated sigma algebra, I only have seen the case that random variable takes discrete values. How should I understand the notion that E[f|g]?