Radon–Nikodym derivative
E59639
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.
Aliases (2)
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 integral
→
Radon–Nikodym theorem → absolute continuity of functions → complex measures → signed measures → |
| requiresProperty |
absolute continuity of measures
→
|
| symbol |
dν/dμ
→
|
| usedFor |
Bayesian statistics
→
Girsanov theorem → 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 → |
Referenced by (5)
| Subject (surface form when different) | Predicate |
|---|---|
|
Lebesgue integration
("Radon–Nikodym theorem")
→
Radon–Nikodym derivative ("Radon–Nikodym theorem") → |
relatedTo |
|
Girsanov theorem
→
|
coreConcept |
|
Cameron–Martin theorem
→
|
involves |
|
Radon–Nikodym derivative
("Lebesgue decomposition of measures")
→
|
usedFor |