Книга: Radon?Nikodym theorem: Mathematics, Functional Analysis, Sigma-Algebra, ?-Finite Measure, Absolute Continuity, Measurable Function, Complex Measure, Jacobian Matrix and Determinant, Probability Theory

High Quality Content by WIKIPEDIA articles! In mathematics, the Radon?Nikodym theorem is a result in functional analysis that states that, given a measurable space (X,?), if a ?-finite measure ? on (X,?) is absolutely continuous with respect to a ?-finite measure ? on (X,?), then there is a measurable function f on X and taking values in [0,?),The function f satisfying the above equality is uniquely defined up to a ?-null set, that is, if g is another function which satisfies the same property, then f = g ?-almost everywhere. f is commonly written d?/d? and is called the Radon?Nikodym derivative. The choice of notation and the name of the function reflects the fact that the function is analogous to a derivative in calculus in the sense that it describes the rate of change of density of one measure with respect to another (the way the Jacobian determinant is used in multivariable integration).