Let $\{X_t\}_t$ be a sequence of discrete random variables and assume that \begin{equation} \\Pr\big(X_t=x \, | \, X_{1},\dots, X_{t-1}\big) \geq \nu_x \quad \text{ for all } t=1,2,\dots \end{equation}

Consider a sequence of binary random variables $\{Y_t\}_t$ such that If $Y_t=1$, then $X_t=x$, i.e., $Y_t\leq 1\{X_t=x\}$, for all $t=1,2,\dots$

Could you help me to show that \begin{equation} \\Pr\big(X_t=x \, | \, Y_{1},\dots, Y_{t-1}\big) \geq \nu_x \quad \text{ for all } t=1,2,\dots \quad ? \end{equation}

Let $\{X_t\}_t$ be a sequence of discrete random variables and assume that \begin{equation} \\Pr\big(X_t=x \, | \, X_{1},\dots, X_{t-1}\big) \geq \nu_x \quad \text{ for all } t=1,2,\dots \end{equation} where $\nu_x\in [0,1]$.

Consider a sequence of binary random variables $\{Y_t\}_t$ such that If $Y_t=1$, then $X_t=x$, i.e., $Y_t\leq 1\{X_t=x\}$, for all $t=1,2,\dots$

Could you help me to show that \begin{equation} \\Pr\big(X_t=x \, | \, Y_{1},\dots, Y_{t-1}\big) \geq \nu_x \quad \text{ for all } t=1,2,\dots \quad ? \end{equation}