Since $E[E[X\vert \mathcal{A}]1_{A}]=E[X1_{A}]$ does $E[E[X\vert \mathcal{A}]]=E[X]$ always hold?

Question

I have shown for a measurable $A$ that $E[E[X\vert \mathcal{A}]1_{A}]=E[X1_{A}]$

Does this mean $E[E[X\vert \mathcal{A}]]=E[X]$ always holds?

since $E[E[X\vert \mathcal{A}]]=E[E[X\vert \mathcal{A}]1_{\Omega}]=E[X1_{\Omega}]=E[X]$, where $\Omega$ is the actual outcome space.

In other words, if the expectation of conditional expectation always expectation itself?

Answer 1

Yes, this is a true statement, and as you note it is an immediate consequence of the definition of conditional expectation.