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. Definition. Let formula_1, formula_2 be two measurable spaces. A function is called a (transition) kernel from formula_4 to formula_5 if the following two conditions hold: Classification of transition kernels. Transition kernels are usually classified by the measures they define. Those measures are defined as with for all formula_6 and all formula_9. With this notation, the kernel  formula_16 is called Operations. In this section, let formula_1, formula_2 and formula_31 be measurable spaces and denote the product σ-algebra of formula_32 and formula_33 with formula_34 Product of kernels. Definition. Let formula_35 be a s-finite kernel from formula_4 to formula_5 and formula_38 be a s-finite kernel from formula_39 to formula_40. Then the product formula_41 of the two kernels is defined as for all formula_44. Properties and comments. The product of two kernels is a kernel from formula_4 to formula_46. It is again a s-finite kernel and is a formula_20-finite kernel if formula_35 and formula_38 are  formula_20-finite kernels. The product of kernels is also associative, meaning it satisfies for any three suitable s-finite kernels formula_52. The product is also well-defined if formula_38 is a kernel from formula_5 to formula_40. In this case, it is treated like a kernel from formula_39 to formula_40 that is independent of formula_4. This is equivalent to setting for all formula_60 and all formula_9. Composition of kernels. Definition. Let formula_35 be a s-finite kernel from formula_4 to formula_5 and formula_38 a s-finite kernel from formula_66 to formula_40. Then the composition formula_68 of the two kernels is defined  as for all formula_9 and all formula_72. Properties and comments. The composition is a kernel from formula_4 to formula_40 that is again s-finite. The composition of kernels is associative, meaning it satisfies for any three suitable s-finite kernels formula_52. Just like the product of kernels, the composition is also well-defined if formula_38 is a kernel from formula_5 to formula_40. An alternative notation is for the composition is formula_80 Kernels as operators. Let formula_81 be the set of positive measurable functions on formula_82. Every kernel formula_16 from formula_4 to formula_5 can be associated with a linear operator given  by The composition of these operators is compatible with the composition of kernels, meaning