More about the tower property of conditional expectation

Question

If filtrations $\mathcal F_i$ i=1,2 don't have the inclusion relation. i.e. neither $\mathcal F_1$ $\subset$ $\mathcal F_2$ nor $\mathcal F_2$ $\subset$ $\mathcal F_1$. What is E[E[X|$\mathcal F_1$]|$\mathcal F_2$] and E[E[X|$\mathcal F_2$]|$\mathcal F_1$]? I try to understand it in an intuitive way that they're projections to the space $\mathcal L^2(\mathcal F_1)$ and $\mathcal L^2(\mathcal F_2)$ .Since they are in different space, they are not the same in general. Do they have relations any more?or Can they be expressed in a different way?

Answer 1

Speaking about projections... Consider the orthogonal projection $p_1$ on the line $y=0$ and the orthogonal projection $p_2$ on the line $x=y$ in the $xy$-plane. Then, for every $(x,y)$ in the plane, $$ p_1(x,y)=(x,0),\qquad p_2(x,y)=\left(\frac{x+y}2,\frac{x+y}2\right), $$ hence $$ p_1(p_2(x,y))=\left(\frac{x+y}2,0\right),\qquad p_2(p_1(x,y))=\left(\frac{x}2,\frac{x}2\right). $$ Do you see any relation between $p_1(p_2(x,y))$ and $p_2(p_1(x,y))$? Me neither.