Transition kernel

In the mathematics of probability, a transition kernel or kernel is a function in mathematics that has different applications. Kernels can for example be used to define random measures or stochastic processes. The most important example of kernels are the Markov kernels. Let , be two measurable spaces. A function is called a (transition) kernel from to if the following two conditions hold: For any fixed , the mapping is -measurable; For every fixed , the mapping is a measure on . Transition kernels are usually classified by the measures they define. Those measures are defined as with for all and all . With this notation, the kernel is called a substochastic kernel, sub-probability kernel or a sub-Markov kernel if all are sub-probability measures a Markov kernel, stochastic kernel or probability kernel if all are probability measures a finite kernel if all are finite measures a -finite kernel if all are -finite measures a s-finite kernel is a kernel that can be written as a countable sum of finite kernels a uniformly -finite kernel if there are at most countably many measurable sets in with for all and all . In this section, let , and be measurable spaces and denote the product σ-algebra of and with Let be a s-finite kernel from to and be a s-finite kernel from to . Then the product of the two kernels is defined as for all . The product of two kernels is a kernel from to . It is again a s-finite kernel and is a -finite kernel if and are -finite kernels. The product of kernels is also associative, meaning it satisfies for any three suitable s-finite kernels . The product is also well-defined if is a kernel from to . In this case, it is treated like a kernel from to that is independent of . This is equivalent to setting for all and all . Let be a s-finite kernel from to and a s-finite kernel from to . Then the composition of the two kernels is defined as for all and all . The composition is a kernel from to that is again s-finite. The composition of kernels is associative, meaning it satisfies for any three suitable s-finite kernels .