Transition kernel

In this section, let ${\displaystyle (S,{\mathcal {S}})}$, ${\displaystyle (T,{\mathcal {T}})}$ and ${\displaystyle (U,{\mathcal {U}})}$ be measurable spaces and denote the product σ-algebra of ${\displaystyle {\mathcal {S}}}$ and ${\displaystyle {\mathcal {T}}}$ with ${\displaystyle {\mathcal {S}}\otimes {\mathcal {T}}}$

Let ${\displaystyle \kappa ^{1}}$ be a s-finite kernel from ${\displaystyle S}$ to ${\displaystyle T}$ and ${\displaystyle \kappa ^{2}}$ be a s-finite kernel from ${\displaystyle S\times T}$ to ${\displaystyle U}$. Then the product ${\displaystyle \kappa ^{1}\otimes \kappa ^{2}}$ of the two kernels is defined as^[3][4]

${\displaystyle \kappa ^{1}\otimes \kappa ^{2}\colon S\times ({\mathcal {T}}\otimes {\mathcal {U}})\to [0,\infty ]}$

${\displaystyle \kappa ^{1}\otimes \kappa ^{2}(s,A)=\int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{A}(t,u)}$

for all ${\displaystyle A\in {\mathcal {T}}\otimes {\mathcal {U}}}$.

The product of two kernels is a kernel from ${\displaystyle S}$ to ${\displaystyle T\times U}$. It is again a s-finite kernel and is a ${\displaystyle \sigma }$-finite kernel if ${\displaystyle \kappa ^{1}}$ and ${\displaystyle \kappa ^{2}}$ are ${\displaystyle \sigma }$-finite kernels. The product of kernels is also associative, meaning it satisfies

${\displaystyle (\kappa ^{1}\otimes \kappa ^{2})\otimes \kappa ^{3}=\kappa ^{1}\otimes (\kappa ^{2}\otimes \kappa ^{3})}$

for any three suitable s-finite kernels ${\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}}$.

The product is also well-defined if ${\displaystyle \kappa ^{2}}$ is a kernel from ${\displaystyle T}$ to ${\displaystyle U}$. In this case, it is treated like a kernel from ${\displaystyle S\times T}$ to ${\displaystyle U}$ that is independent of ${\displaystyle S}$. This is equivalent to setting

${\displaystyle \kappa ((s,t),A):=\kappa (t,A)}$

for all ${\displaystyle A\in {\mathcal {U}}}$ and all ${\displaystyle s\in S}$.^[4][3]

Let ${\displaystyle \kappa ^{1}}$ be a s-finite kernel from ${\displaystyle S}$ to ${\displaystyle T}$ and ${\displaystyle \kappa ^{2}}$ a s-finite kernel from ${\displaystyle S\times T}$ to ${\displaystyle U}$. Then the composition ${\displaystyle \kappa ^{1}\cdot \kappa ^{2}}$ of the two kernels is defined as^[5][3]

${\displaystyle \kappa ^{1}\cdot \kappa ^{2}\colon S\times {\mathcal {U}}\to [0,\infty ]}$

${\displaystyle (s,B)\mapsto \int _{T}\kappa ^{1}(s,\mathrm {d} t)\int _{U}\kappa ^{2}((s,t),\mathrm {d} u)\mathbf {1} _{B}(u)}$

for all ${\displaystyle s\in S}$ and all ${\displaystyle B\in {\mathcal {U}}}$.

The composition is a kernel from ${\displaystyle S}$ to ${\displaystyle U}$ that is again s-finite. The composition of kernels is associative, meaning it satisfies

${\displaystyle (\kappa ^{1}\cdot \kappa ^{2})\cdot \kappa ^{3}=\kappa ^{1}\cdot (\kappa ^{2}\cdot \kappa ^{3})}$

for any three suitable s-finite kernels ${\displaystyle \kappa ^{1},\kappa ^{2},\kappa ^{3}}$. Just like the product of kernels, the composition is also well-defined if ${\displaystyle \kappa ^{2}}$ is a kernel from ${\displaystyle T}$ to ${\displaystyle U}$.

An alternative notation is for the composition is ${\displaystyle \kappa ^{1}\kappa ^{2}}$^[3]

Let ${\displaystyle {\mathcal {T}}^{+},{\mathcal {S}}^{+}}$ be the set of positive measurable functions on ${\displaystyle (S,{\mathcal {S}}),(T,{\mathcal {T}})}$.

Every kernel ${\displaystyle \kappa }$ from ${\displaystyle S}$ to ${\displaystyle T}$ can be associated with a linear operator

${\displaystyle A_{\kappa }\colon {\mathcal {T}}^{+}\to {\mathcal {S}}^{+}}$

given by^[6]

${\displaystyle (A_{\kappa }f)(s)=\int _{T}\kappa (s,\mathrm {d} t)\;f(t).}$

The composition of these operators is compatible with the composition of kernels, meaning^[3]

${\displaystyle A_{\kappa ^{1}}A_{\kappa ^{2}}=A_{\kappa ^{1}\cdot \kappa ^{2}}}$