Any equivalent process to a right continuous process has a right continuous modification Let $Y$ be an equivalent process, i.e. has the same finite dimensional distributions of $X$. Then if $X$ is right continuous, then there is a modification of $Y$ that is right continuous. 
This is a problem from Ethier and Kurtz Markov Processes. This problem is given in the early part of the chapter after introductions to definitions on stochastic processes. I can't think of any way to prove this fact using definitions. I would greatly appreciate any help.
 A: I assume (as in Ethier-Kurtz) that $X$ and $Y$ take values in a metric space $E$. In fact, to be able to apply a known result later, I suppose that $E$ is complete and separable. Let $\Bbb Q_+$ denote the non-negative rational numbers, and let $H:=E^{\Bbb Q_+}$ denote the space of paths from $\Bbb Q_+$ to $E$ with associated product $\sigma$-algebra $\mathcal H:=\mathcal E^{\otimes\Bbb Q_+}$. (Here $\mathcal E$ is the Borel $\sigma$-algebra on $E$.) Let $(\Omega_1,\mathcal F_1,P_1)$ denote the probability space on which $Y$ is defined. Then $Y$, viewed as a map $\omega\mapsto(Y_s(\omega):s\in\Bbb Q)$ from $\Omega_1$ to $H$, induces a probability measure $P_Y$ on $(H,\mathcal H)$: $P_Y(B)=P_1(\{\omega\in\Omega_1: Y(\omega)\in B\})$, $B\in\mathcal H$.
Likewise, $X$ induces $P_X$, and $P_Y=P_X$ because $X$ and $Y$ are equivalent.
Now define $G:=\{x\in H: x$ is the restriction to $\Bbb Q$ of a right-continuous map of $[0,\infty)$ into $E\}$. It is known that $G$ is in the completion of $\mathcal H$ for each probability measure on $(H,\mathcal H)$ (see, for example, Theorem IV-18 in volume A of Probabilities and Potential by Dellacherie and Meyer), so writing $\overline P_X$ for the completion of $P_X$, one checks that $\overline P_X(G)=1$ because $X$ has right-continuous paths. It follows that $\overline P_Y(G)=1$ as well. Consequently there exists $G_0\in\mathcal H$ with $G_0\subset G$ and $P_Y(G_0)=1$.
Finally, define $Z_t(\omega):=\lim_{s\downarrow t,s\in\Bbb Q}Y_s(\omega)$, $t\ge 0$, if $\omega\in Y^{-1}(G_0)$ and $Z_t(\omega)=e_0$ for all $t\ge 0$ if $\omega\in Y^{-1}(G_0)$, where $e_0$ is a fixed point of $E$.
(I suspect this is not the argument envisioned by Ethier-Kurtz for an exercise appearing so early in their treatment, but a simpler proof does not occur to me.)
