Existence of something like regular conditional distribution

Question

This exercise is from Durrett's Probability: Theory and Examples, Exercise 5.1.16.

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Suppose $X$ and $Y$ take values in a nice space $(S,\mathcal{S})$ and $\mathcal{G}=\sigma(Y).$ Then prove that there is a function $\mu:S\times S\to[0,1]$ so that

1. For each $A$, $\mu(Y(\omega),A)$ is a version of $\mathbf{P}(X\in A|\mathcal{G})$

2. For a.e. $\omega,$ $A\to \mu(Y(\omega),A)$ is a probability measure on $(S,\mathcal{S}).$

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My attempt: Initially, I thought that the function $(\omega,A)\mapsto \mu(Y(\omega),A)$ trivially satisfies the condition (i) and (ii) if $\mu$ is a regular conditional probability. However, I found that is wrong since if $\omega_0\in S$ is an element such that $A\to \mu(\omega_0,A)$ is NOT a probability measure and $Y$ is a constant random variable such that $Y(\omega)=\omega_0$ for all $\omega \in S$, then my assertion fails. How should I modify the argument here?

Thanks in advance!