Random measure

In probability theory, a random measure is a measure-valued random element. Let X be a complete separable metric space and ${\displaystyle {\mathfrak {B}}(X)}$ the σ-algebra of its Borel sets. A Borel measure μ on X is boundedly finite if μ(A) < ∞ for every bounded Borel set A. Let ${\displaystyle M_{X}}$ be the space of all boundedly finite measures on ${\displaystyle {\mathfrak {B}}(X)}$. Let (Ω, ℱ, P) be a probability space, then a random measure maps from this probability space to the measurable space (${\displaystyle M_{X}}$, ${\displaystyle {\mathfrak {B}}(M_{X})}$).A measure generally might be decomposed as:

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