1. The cost function to be minimized $c(x, u)$ is bounded and continuous on both $\mathbb{U}$ and $\mathbb{X}$ 2. If applicable, $c_{N}$ is bounded and continuous;
3. $\mathbb{U}_{t}(x) = \mathbb{U}$ is compact; and 4. $\mathcal{T}(\cdot\mid x,u)$ is Feller Property.

1. For every $x\in\mathbb{X}$ the bounded measurable cost function $c(x, u)$ is continuous on $\mathbb{U}$ 2. If applicable, $c_{N}$ is bounded and measurable; 3. $\mathbb{U}_{t}(x) = \mathbb{U}$ is compact; 4. $\mathcal{T}(\cdot|x,u)$ is Feller Property in $u$ for fixed $x$.