Suppose we have a stochastic series $\{X_t\in\mathbb{R}, t=1,\cdots, T\}$. Further suppose that $G(X_t)=\mathbf{1}_{X_t\geq 0}$ where $\mathbf{1}$ is an indicator function. Can it be concluded that the conditional density
\begin{equation} f_{X_t\mid G(X_t)}(x_t) = f_{X_t}(x_t) \end{equation}
I think not, as the function alone suggests dependence between $X_t$ and $G(X_t)$. If that is not the case, assuming that $X_t\sim N(0,\sigma^2)$ for all $t$, how can we express/ evaluate the conditional density function on the left hand side of the above equation?
