This is a very basic question in the theory of Markov chains and I'm just not sure how to prove it mathematically.
Say we have random variables $X, Y$ that are correlated and we have a possibly random function of the random variable $Y$ denoted $g(Y)=\hat{X}$. Is it obvious that $X\rightarrow Y \rightarrow \hat{X}$? How does one give a sound proof of this?
This is from Tom Cover's "Elements of Information Theory", where Fano's inequality gets proven.