Under what conditions does a specified conditional distribution exist It is common to see conditional distributions specified in stats like:
$$(X \mid \mu = t) \sim \mathcal{N} (t, 1)$$
(Of course, we can also use some other distribution here)
How do you prove that such a conditional probability actually exists, in terms of a regular conditional probability? And is there some condition on the underlying probability space?
Proofs and/or references appreciated.

On a polish space $(\Omega,\mathcal{F},P)$, a regular conditional distribution exists, that is, given any $\mathcal{G}$ sub-sigma algebra there is a function $K: \Omega\times \mathcal{F} \to [0,1]$ such that
1) $K(\omega, \cdot)$ is a probability measure
2) $K(\cdot, A)$ is $\mathcal{G}$ measurable
3) $K(\cdot, A) = \Bbb{E}[1_A \mid \mathcal{G}]$ $P$ almost surely
Moreover, if $\mathcal{H}\subset \mathcal{G}$ is countably generated ($\mathcal{H} = \sigma(H_n, n \in \Bbb{N})$) then the following substitution principle holds:
4) There is a null set $N\in \mathcal{G}$: 
$$K(\omega,A) = 1_A(\omega) \quad A \in \mathcal{H}, \omega\in \Omega \setminus N$$
An interesting case occurs when $\mu$ is a random variable taking values on $\Bbb{R}$ since $\sigma(\mu)$ is countably generated we have that
$$K(\omega,\{\omega'\mid \{\mu(\omega') = \mu(\omega)\}) = 1, \quad \omega\in \Omega \setminus N$$
So almost surely $K(\omega, \cdot)$ is concentrated on the level set $\{\mu(\omega') = \mu(\omega)\}$
In the case you are dealing with, you know further (by assumption) that if $\mu(\omega )= t$
$$K(\omega, A) = \int_A \frac{1}{\sqrt{2\pi}}\exp\big(\frac{(x-t)^2}{2}\big)\, dx$$
that is, $K(\omega, \cdot) \sim N(\mu(\omega, 1))$ almost surely.
References for this can be found in 
Parthasarathy, K. R., Probability Measures on Metric Spaces 1967.
Karatzas and Schreve - Brownian motion and stochastic calculus [chap 5, pag 306] 1991
and in the notes pointed out by Calculon:Theorem B.18 in the Appendix of the following lecture notes math.wisc.edu/~seppalai/courses/735/notes.pdf
