Let $(\Omega,\mathcal F,\{\mathcal F_t\}_{t\in [0;T]},\mathbb P)$ be a filtered probability space satisfying the usual conditions and let $\{X_t\}_{t\in[0;T]}$ be an adapted stochastic process, whose paths are right-continuous step functions with finitely many steps. As usual, $\Delta X_t := X_t - X_{t-}$ and write $X_{0-} := 0$. Define $$ Y_t := \sum_{s \in [0;t], \Delta X_s > 0} \Delta X_s. $$
Is $Y$ necessarily adapted and how can I prove that?