The theorem is concerned with the existence of density (derivative) of one measure with respect to another. Let \((\Omega,\mathcal{F})\) be a measurable space, i.e., a set Ω together with a σ-algebra \(\mathcal{F}\) of subsets of Ω. Suppose that ν, μ are two σ-finite positive measures on \((\Omega,\mathcal{F})\) such that ν is absolutely continuous (denoted by ν ≪ μ) with respect to μ, i.e., if μ(A) = 0 for some \(A \in \mathcal{F}\) then ν(A) = 0. The Radon–Nikodým theorem states that these exists a μ-integrable function f : Ω → ℝ + such that
Moreover, f is μ-a.e. unique, in the sense that if f′ also satisfies the above then the μ-measure of the points ω such that f(ω)≠f′(ω) equals zero. The function f is called Radon–Nikodým derivative of μ with respect to ν and this is often denoted by
The standard proof is as follows. First, assume that μ(Ω)...
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References and Further Reading
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Konstantopoulos, T., Zerakidze, Z., Sokhadze, G. (2011). Radon–Nikodým Theorem. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_468
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