Assumption (1)

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 Weak Feller.

Assumption (2)

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 Strong Feller in $u$ for fixed $x$.