Properties of Expectation Conditioned on a Function of Random Variables

Question

I've come across the two following claims regarding the properties of conditional expectation:

$$ \mathsf{E} \Big\{ \mathsf{E}\big[Y\, \big|\, F(X) \big]\; \Big|\; X \Big\} = \mathsf{E}\big[Y\, \big|\, F(X) \big] \tag{1} $$

and

$$ \mathsf{E} \Big\{ \mathsf{E}\big[Y\, \big|\, X \big]\; \Big|\; F(X) \Big\} = \mathsf{E}\big[Y\, \big|\, F(X) \big]\,, \tag{2} $$

where $F(\cdot)$ is a given function on the range of $X.$

How does one show this?

Answer 1

Observe that

$$ \sigma \big(F(X)\big) \subseteq \sigma(X)\,. $$

Therefore, (1) and (2), written above, follow immediately from the tower property.