Random measure

In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. Random measures can be defined as transition kernels or as random elements. Both definitions are equivalent. For the definitions, let E {displaystyle E} be a separable complete metric space and let E {displaystyle {mathcal {E}}} be its Borel σ {displaystyle sigma } -algebra. (The most common example of a separable complete metric space is R n {displaystyle mathbb {R} ^{n}} ) A random measure ζ {displaystyle zeta } is a (a.s.) locally finite transition kernel from a (abstract) probability space ( Ω , A , P ) {displaystyle (Omega ,{mathcal {A}},P)} to ( E , E ) {displaystyle (E,{mathcal {E}})} .