When studying Markov processes, I have seen a lot of authors define the semigroup as $P_tf(x) = \mathbb E_x(f(X_t))$ (with the assumption that $X_t$ is homogeneous) and the call $\mathbb E_x$ as the "expectation given $X_0=x$", i.e, they mean
$$\mathbb E_x(\cdot) = \mathbb E[\cdot|X_0=x],$$
and I couldn't find a rigorous definition of this because if $X_0$ is an absolute continuous variable then the right-hand side wouldn't work in the usual sense (dividing $\mathbb P(X_0=x)$). However, I also notice that some authors avoid defining this conditional law by starting out with "Markov kernels" associated with a Markov process $(X_t)$, which totally makes sense to me. I'm okay with the latter approach although there are things that I'm not fully understanding right now but I will reserve it for another post.
In addition, some even define $X=(X_t)_{t\geq 0}$ to be homegeneous iff for every bounded measurable set $\Gamma$ (in a metric space where $X_t$ takes value) we have
$$ \mathbb P(X_t \in \Gamma | X_s) = \mathbb P (X_{t+u} \in \Gamma|X_{s+u}), \quad \forall u >0. $$
First question: Without mentioning anything else, should I think interpret this equality as almost surely and the left handside is $\sigma(X_{s+u})$ measurable and is a version of $\mathbb P (X_{t+u} \in \Gamma|X_{s+u})$?
Second question: I'm looking for the rigorous definition of $\mathbb E[\cdot|X_0=x]$ above, I believe that it should take a deterministic value for $P_tf(x)$ to make sense.
Any rigorous reference related to this is highly appreciated. Thank you for your help!