Consider a probability density function $f(x, y)$ on $\mathbb{R}^2$. The probability space is $(\mathbb{R}^2, \mathcal{B}, f \, d\lambda)$, where $\mathcal{B}$ is the Borel $\sigma$-algebra on $\mathbb{R}^2$, and $f \, d\lambda$ denotes the probability measure with density $f(x, y)$.
Let $\mathcal{C} = \{Z \times \mathbb{R} : Z \text{ is a Borel subset of } \mathbb{R}\}$.
Show that the regular conditional probability $Q^{\mathcal{C}}(M \mid (x, y))$ is given by:
$$ Q^{\mathcal{C}}(M \mid (x, y)) = \frac{\int_{M} f(x, y) \, dy}{\int_{\mathbb{R}^2} f(x, y) \, dy}. $$
Can someone tell me how to show this? I have only been given this definition: Let $(Ω, \mathcal{F}, P)$ be a probability space, $\mathcal{B} \subset \mathcal{F}, Q^\mathcal{B}(A, \omega) : \mathcal{F} \times \Omega \to [0,1]$ is a regualar conditional probability on $\mathcal{F}$ given $\mathcal{B}$ under P if $(i) \forall A \in \mathcal{F}$, it is a version of $E(\mathbb{1_A})$ $(ii) \forall \omega \in \Omega$,as a function of A it is a probability on $(\Omega, \mathcal{F})$.
