Product Spaces, Transition Kernel and Rubini's Theorem

Lemma 1. Sectioning sets Sections of measurable sets are measurable.  If \(A\in \mathcal{B}_1 \times \mathcal{B}_2\), then for all \(\omega_1 \in \Omega_1\),
\[A_{w_1}\in \mathcal{B_2}\]
Corollary 1. Sections of measurable functions are measurable.  That is if
\[ X: (\Omega_1\times \Omega_2, \mathcal{B}_1 \times \mathcal{B}_2) \mapsto (S,\mathcal{S})\] then \[ X_{\omega_1} \in \mathcal{B}_2. \]  We say \(X_{\omega_1}\) is \(\mathcal{B}/\mathcal{S}\) measurable.

Define the transition (probability) kernel

\[K(\omega_1,A_2):\Omega_1 \times \mathcal{B}_2 \mapsto [0,1]\]

if it satisfies the following

1. for each \(\omega_1, K(\omega_1,\cdot)\) is a probability measure on \(\mathcal{B}_2\), and
2. for each \(A_2\in \mathcal{B}_2, K(\cdot, A_2)\) is \(\mathcal{B}_1/\mathcal{B}([0,1])\) measurable.

Transition kernels are used to define discrete time Markov processes where \(K(\omega_1,A_2)\) represents the conditional probability that, starting from \(\omega_1\), the next movement of the system results in a state in \(A_2\).

Theorem 1.  Let \(P_1\) be a probability measure on \(\mathcal{B}_1\), and suppose

\[K:\Omega_1 \times \mathcal{B}_2 \mapsto [0,1]\] is a transition kernel.  Then \(K\) and \(P_1\) uniquely determine a probability on \(\mathcal{B}_1 \times \mathcal{B}_2\) via the formula

\[P(A_1\times A_2)= \int_{A_1}K(\omega_1,A_2)P_1(dw_1),\] for all \(A_1\times A_2\) in the class of measurable rectangles.

Theorem 2. Marginalization  Let \(P_1\) be a probability measure on \((\Omega_1,\mathcal{B}_1)\) and suppose \(K: \Omega_1\times \mathcal{B_2} \mapsto [0,1]\) is a transition kernel.  Define \(P\) on \((\Omega_1 \times \Omega_2, \mathcal{B_1}\times \mathcal{B_2})\) by
\[P(A_1\times A_2)= \int_{A_1} K(\omega_1,A_2)P_1(d\omega_1). \]  Assume
\[X:(\Omega_1\times \Omega_2, \mathcal{B}_1\times \mathcal{B}_2) \mapsto (\mathbb{R},\mathcal{B}(\mathbb{R})) \] and furthermore suppose \(X\) is integrable.  Then
\[Y(\omega_1)=\int_{\Omega_2} K(\omega_1,d\omega_2)X_{\omega_2}(\omega_2)\] has the properties

1. \(Y\) is well defined.
2. \(Y \in B_1\)
3. \(Y \in L_1(P_1)\) and furthermore

\[\int_{\Omega_1\times \Omega_2}XdP = \int_{\Omega_1} Y(\omega_1)P_1(d\omega_1) = \int_{\Omega_1} [ \int_{\Omega_2} K(\omega_1,d\omega_2) X_{\omega_1}(d\omega_2)]P_1(\omega_1). \]

Theorem 3.  Fubini Theorem  Let \(P=P_1\times P_2\) be a product measure.  If \(X\) is \(\mathcal{B}_1 \times \mathcal{B}_2\) measurable and is either non-negative or integrable with respect to \(P\) then

\[\begin{aligned}
\int_{\Omega_1\times \Omega_2}XdP &= \int_{\Omega_1}[ \int_{\Omega_2}X_{\omega_1}(\omega_2) P_2(d\omega_2) ] P_1(d\omega_1) \\
                                                                &= \int_{\Omega_2}[ \int_{\Omega_1}X_{\omega_2}(\omega_1) P_1(d\omega_1) ] P_2(d\omega_2)
\end{aligned} \]