Let $f: \mathbb{R}^n \times \omega \to \mathbb{R}$ where $\omega \sim D$. Assume $(f(x_k, \omega_k))$ be a random sequence such that
$$ \mathbb{E}_{\omega^{k+1}} [f(x_{k+1}, \omega_{k+1})] \leq f(x_{k}, \omega_{k}) $$ where $f(x_{k}, \omega_{k})$ is just a realization at the $k$-th step.
Question1
Is $(f(x_k, \omega_k))$ a supermartingale sequence?
Question2
If so, assuming $f$ is bounded below, can we show that $(f(x_k, \omega_k))$ converges to some random variable with probability one using Doob's martingale convergence theorems?