Is the following statement true?
$\sum_{j}Pr(a|b=b_j)Pr(b=b_j|c=c^*)=Pr(a|c=c^*)$
I'm using this statement (with a simplification that g takes on a finite number of values) to convince myself that
$E[E(u_i|\mathbf{g_{i-1},...,g_1})|u_{i-1},...,u_1]=E(u_i|u_{i-1},...,u_1)$ where $\bf{g_i}=\bf{x_i}$$u_i$ and $\mathbf{x_i}$ is a k-dimensional vector.
(by the Law of Iterated Expectations)
The nesting of information sets isn't very clear to me here so I've been trying to convince myself of it but I'm not sure if I'm doing it right.
It is not, thought it is close to the correct formula $$\sum_j P(a|\{b=b_j\}\cap \{c=c^*\})P(b=b_j|c=c^*)=P(a|c=c^*).$$ To see this, let $B_j$ be the event $b=b_j$, and $C$ be the event $c=c^*$. Then $$ P(a|C) =\frac{P(a\cap C)}{P(C)} =\frac{\sum_j P(a\cap B_j\cap C)}{P(C)} =\sum_j \frac{P(a\cap B_j\cap C)}{P(B_j\cap C)}\cdot\frac{P(B_j\cap C)}{P(C)}\\ \qquad\qquad\qquad\qquad\qquad\qquad\,\,\,\,\,\,=\sum_jP(a|B_j\cap C)\cdot P(B_j|C). $$