Radon–Nikodym derivative

E59639

mathematical concept measure-theoretic notion

The Radon–Nikodym derivative is a function that represents how one measure changes with respect to another absolutely continuous measure, playing a central role in modern probability theory and measure theory.

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All labels observed (3)

Label	Occurrences
Radon–Nikodym theorem	7
Radon–Nikodym derivative canonical	4
Lebesgue decomposition of measures	1

Statements (41)

Predicate	Object
instanceOf	mathematical concept ⓘ measure-theoretic notion ⓘ
appliesTo	finite measures ⓘ σ-finite measures ⓘ
assumes	complete measure space (often) ⓘ underlying σ-algebra ⓘ
codomain	extended real-valued functions ⓘ
condition	ν is absolutely continuous with respect to μ ⓘ
describes	rate of change of one measure with respect to another ⓘ
domain	measure space ⓘ
expresses	ν(A) = ∫_A (dν/dμ) dμ for measurable sets A ⓘ
field	measure theory ⓘ probability theory ⓘ
generalizes	classical derivative of distribution functions ⓘ density of a probability distribution ⓘ
mathematicalNature	measurable function ⓘ
namedAfter	Johann Radon ⓘ Otton Nikodym ⓘ
property	linearity in the measure ⓘ non-negativity when measures are positive ⓘ uniqueness up to μ-almost everywhere equality ⓘ
relatedTo	Lebesgue integration ⓘ surface form: Lebesgue integral Radon–Nikodym derivative self-linksurface differs ⓘ surface form: Radon–Nikodym theorem absolute continuity of functions ⓘ complex measures ⓘ signed measures ⓘ
requiresProperty	absolute continuity of measures ⓘ
symbol	dν/dμ ⓘ
usedFor	Bayesian inference ⓘ surface form: Bayesian statistics Girsanov theorem ⓘ Radon–Nikodym derivative self-linksurface differs ⓘ surface form: Lebesgue decomposition of measures change of measure in probability ⓘ defining conditional expectation ⓘ defining densities of measures ⓘ defining likelihood ratios ⓘ stochastic calculus ⓘ
usedIn	ergodic theory ⓘ financial mathematics ⓘ information theory ⓘ martingale theory ⓘ statistical inference ⓘ