Consider a complete metric compact space $X$. For each $x\in X$ we define a probability measure $T(\cdot|x)$ over a Borel sigma-algebra $\mathcal{B}(X)$. We call a set $A\subset X$ invariant if $T(A|x) = 1$ for all $x\in A$. Does it mean that if $A$ is invariant, the same holds for its closure?
I am especially interested in the case when $T$ is Feller continuous or strong Feller continuous.
More precisely, denote $ \mathcal{P}f(x) = \int\limits_X f(y)T(dy|x) $ and spaces $\mathcal{M}_b$ and $\mathcal{C}_b$ of measurable bounded and continuous bounded functions on $X$. Then Feller continuity means $f(x)\in \mathcal{C}_b \Rightarrow \mathcal{P}f(x)\in\mathcal{C}_b$ and strong Feller continuity means $f(x)\in \mathcal{M}_b \Rightarrow \mathcal{P}f(x)\in\mathcal{C}_b$.