Radon–Nikodym theorem

The Radon–Nikodym theorem states that a σ-finite measure absolutely continuous with respect to another σ-finite measure has a measurable density whose integral recovers the first measure.

Absolute continuity of measures: A measure is absolutely continuous with respect to another if every null set of the latter is also null for the former. This is the theorem’s key hypothesis and the condition that permits a density.
Measure: A countably additive function assigning nonnegative sizes to sets in a σ-algebra. The theorem compares two measures and represents one through the other.
Probability density function: A nonnegative function whose integral over an event gives its probability relative to a reference measure. When a probability law is absolutely continuous, its density is a Radon–Nikodym derivative.
Riesz representation theorem: A family of theorems representing linear functionals as integration against measures or vectors. It represents functionals through measures, while Radon–Nikodym represents one measure through another.
Radon–Nikodym derivative: A measurable function whose integral over each measurable set equals a measure absolutely continuous with respect to a reference measure. It is the density whose existence the theorem guarantees.
σ-algebra: A collection of sets closed under complements and countable unions. Measures and measurable densities are defined relative to σ-algebras.
Likelihood ratio: The ratio of probability densities for two statistical models on the same observation space. It is a Radon–Nikodym derivative when one probability law is absolutely continuous with respect to another.
Fubini's theorem: A theorem allowing integrals on product spaces to be evaluated as iterated integrals under suitable conditions. It rearranges integration; it does not produce a density from absolute continuity.
Lebesgue integral: An integral defined by measuring the sizes of level sets of measurable functions. Integrating the derivative over a set reconstructs the measure assigned to that set.
Measurable function: A function whose inverse images of measurable sets are measurable. The density supplied by the theorem must be measurable.

Linked from 12 pages