Let $B_b(E)$ be the space of bounded measurable functions on some Polish space $E$ endowed with the supremum norm. It seems quite classical that Markov semigroups $P_t:B_b(E)\to B_b(E)$ are in one to one correspondence with Markov processes on $E$.

By Markov semigroup I mean a strongly continuous semigroup $P_t:B_b(E)\to B_b(E)$ satisfying

- $P_t$ is conservative for all $t$, i.e. $P_t1=1$
- $\|P_t\|=1$, for all $t$
- $P_t$ is a positive operator for all $t$

From such an object, the natural way to build the transition functions $p_t(x,dy)$ of an eventually associated Markov process is to set $p_t(x,A):=P_t\mathbb{1}_A(x)$, for all measurable set $A$ and all $x$ in $E$.

However, it does not seem direct that $p_t(x,\cdot)$ is $\sigma$-additive and I was not able to find a proper reference showing this (in the particular context of semigroups on $B_b(E)$).

In fact, the only references I found assume that $P_t$ is bounded pointwise continuous, i.e. for any sequence $(f_n)_n$ of $B_b(E)$ converging pointwisely to a function $f$ and being uniformly bounded, then $$P_t f_n(x)\to P_tf(x)$$ for all $x$ as $n$ goes to infinity. Property which is in fact equivalent to the $\sigma$-additivity of $p_t$.

So, are Markov semigroup bp-continuous ? If yes, how can it be derived from the assumptions? if no, is there a counterexample ?

thanks for your time