What is this conditional expectation equivalent to?

Question

E[E[X|G]|G]

G is a sigma-field and X is G-Measurable.

What is this conditional expectation equivalent to?

Answer 1

In general if random variable $Y$ is $\mathcal G$-measurable then we can take: $$\mathbb E[Y\mid\mathcal G]=Y$$

Applying that on $Y=\mathbb E[X\mid \mathcal G]$ we find:$$\mathbb E[\mathbb E[X\mid \mathcal G]\mid \mathcal G]=\mathbb E[X\mid\mathcal G]$$