I wondering how to prove $X$, $Y$, $g(Y)$ is a Markov Chain in That Order?
$X$, $Y$, $Z$ is a Markov Chain in That Order (denoted $X\to Y\to Z$) if $p(x,y,z) = p(x)\cdot p(y\mid x)\cdot p(z\mid y).$ A property of a Markov Chain is if $X\to Y\to Z$, then $ p(xz\mid y) = p(x\mid y)\cdot p(z\mid y).$
Setting $Z = g(Y)$, I must show: $p(x,y,g(y)) = p(x)\cdot p(y\mid x)\cdot p(g(y)\mid y)$.
I believe I must simplify $p(g(y)\mid y)$ somehow, but I am only aware of conditional expectation identity where $E[g(Y)\mid Y] = g(Y)$ and $E[g(Y)\mid Y=y] = g(y)$.
Thanks for your assistance.