Conditional expectation with respect to union of two sigma-algebra

Question

Let $\mathscr{F}_1$ and $\mathscr{F}_2$ be two independent (sub)sigma-algebra,$X$ be a random variable with $\mathbb{E}X<\infty$.

What we want to find is the relationship between $\mathbb{E}(X|\mathscr{F_1})$,$\mathbb{E}(X|\mathscr{F_2})$, and $\mathbb{E}(X|\sigma(\mathscr{F_1}\cup\mathscr{F_2}))$. Some assumption such as the measurability of $X$ could be added.

Any comment will be appreciated.

Answer 1

For what it's worth: $$\mathbb{E}[X|\mathscr {F_1}](\omega_1)=\int\mathbb{E}[X|\mathscr{F}_1\circ\mathscr{F}_2](\omega_1,\omega_2)dP_\Omega(\omega_2)$$ Where $\mathscr{F}_1\circ\mathscr{F}_2$ is the $\sigma$-algebra associated with the product space $\Omega(\mathscr{F}_1)\times\Omega(\mathscr{F}_2)$.
And where it is also fulfilled that $\mathbb{E}[X|\sigma(\mathscr{F}_1\cup\mathscr{F}_2)](\omega)=\mathbb{E}[X|\mathscr{F}_1\circ\mathscr{F}_2](\omega,\omega)$.