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== Formal definition ==
Let <math>(X,\mathcal A)</math> and <math>(Y,\mathcal B)</math> be [[measurable space]]s. A ''Markov kernel'' with source <math>(X,\mathcal A)</math> and target <math>(Y,\mathcal B)</math>, sometimes written as <math>\kappa:(X,\mathcal{A})\to(Y,\mathcal{B})</math>, is a
# For every (fixed) <math>B_0 \in \mathcal B</math>, the map <math> x \mapsto \kappa(B_0, x)</math> is <math>\mathcal A</math>-[[measurable function|measurable]]
# For every (fixed) <math> x_0 \in X</math>, the map <math> B \mapsto \kappa(B, x_0)</math> is a [[probability measure]] on <math>(Y, \mathcal B)</math>
In other words it associates to each point <math>x \in X</math> a [[probability measure]] <math>\kappa(dy|x): B \mapsto \kappa(B, x)</math> on <math>(Y,\mathcal B)</math> such that, for every measurable set <math>B\in\mathcal B</math>, the map <math>x\mapsto \kappa(B, x)</math> is measurable with respect to the [[Σ-algebra|<math>\sigma</math>-algebra <math>\mathcal A</math>]].<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability Theory: A Comprehensive Course|series=Universitext |year=2014 |publisher=Springer|page=180|edition=2|doi=10.1007/978-1-4471-5361-0|isbn=978-1-4471-5360-3 }}</ref>
== Examples ==
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where <math> x </math> is the number of element at the state <math> n </math>, <math>\xi_i</math> are [[Independent and identically distributed random variables|i.i.d.]] [[random variable]]s (usually with mean 0) and where <math>\mathbf{1}_B</math> is the indicator function. For the simple case of [[Bernoulli distribution|coin flips]] this models the different levels of a [[Galton board]].
== Composition of Markov Kernels
Given measurable spaces <math>(X, \mathcal A)</math>, <math>(Y, \mathcal B) </math> we consider a Markov kernel <math>\kappa: \mathcal B \times X \to [0,1]</math> as a morphism <math>\kappa: X \to Y</math>. Intuitively, rather than assigning to each <math>x \in X</math> a sharply defined point <math> y \in Y</math> the kernel assigns a "fuzzy" point in <math>Y</math> which is only known with some level of uncertainty, much like actual physical measurements. If we have a third measurable space <math>(Z, \mathcal C)</math>, and probability kernels <math>\kappa: X \to Y</math> and <math>\lambda: Y \to Z</math>, we can define a composition <math>\lambda \circ \kappa : X \to Z</math> by the [[Chapman-Kolmogorov equation]]
:<math>(\lambda \circ \kappa) (dz|x) = \int_Y \lambda(dz | y)\kappa(dy|x)</math>.
The composition is associative by the Monotone Convergence Theorem and the identity function considered as a Markov kernel (i.e. the delta measure <math> \kappa_{1}(dx'|x) = \delta_x(dx')</math>) is the unit for this composition.
This composition defines the structure of a [[category (mathematics)|category]] on the measurable spaces with Markov kernels as morphisms, first defined by Lawvere
== Probability Space defined by Probability Distribution and a Markov Kernel==
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* {{citation|first1=Heinz|last1=Bauer|title=Probability Theory|publisher=de Gruyter|year=1996|isbn=3-11-013935-9}}
:: §36. Kernels and semigroups of kernels
== See also ==
* [[Category of Markov kernels]]
[[Category:Markov processes]]
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