Let $(\Omega,\Sigma, P)$ be a probability space where $\Omega$ is a (possibly infinite) state space and $\Sigma$ is the $\sigma$ algebra. Let $S$ be a finite set, and $f: \Omega\to \Delta(S)$ be a $\Sigma$-measurable function, where $\Delta(S)$ denotes all probability distributions over $S$.
Then $P$ and $f$ define a probability measure $Q\in \Delta(S\times \Omega)$ by $$dQ(s,\omega)=f(s|\omega)dP(\omega)\tag{*}$$
Conversely, for any $Q$, I think we can define the corresponding $f$ and $P$.
My question is the following: is the above notation $(*)$ legitimate? What would be the standard notation for $dQ(a,\omega)$ in measure theory?
For another question, suppose $\Omega=\Omega_1\times \Omega_2$, for integration purpose, I need to define the conditional probabilty measure for $\omega_1$ given any $\omega_2$ by something like
$$dP(\omega_1|\omega_2)=P(\tilde{\omega}_1=\omega_1|\tilde{\omega}_2=\omega_2) \tag{**}$$
What would be the standard notation for $dP(\omega_1|\omega_2)$?
Thanks.