A charge that is absolutely continuous with respect to some measure has a density with respect to that is summable with respect to this measure. It was established by J. Radon  and O.M. Nikodým . More precisely, on a measurable space , where is a -algebra of subsets of , suppose one is given a charge , i.e. a countably-additive real or complex function given on , and a -finite measure , and, moreover, let be absolutely continuous with respect to . Then there is a function , , summable with respect to , such that for any set ,
The function is unique (except for modifications on a set of -measure zero), and is called the density of the charge with respect to the measure . There are (see ) generalizations of the theorem to the case when the charge takes values in some vector space.
|||J. Radon, "Ueber lineare Funktionaltransformationen und Funktionalgleichungen" Sitzungsber. Acad. Wiss. Wien , 128 (1919) pp. 1083–1121|
|||O.M. Nikodým, "Sur une généralisation des intégrales de M. J. Radon" Fund. Math. , 15 (1930) pp. 131–179|
|||N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)|
|||J. Diestel, J.J. Uhl jr., "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977)|
The notion of "charge" is not well established in the West; one usually says "signed measure" (cf. Charge). The density is also well defined if is the sum of a series of (non-negative) measures; in this case and the integral may take the value .
The theorem is false if fails to satisfy same finiteness condition; see [a1], §19, for a thorough discussion and illuminating examples.
For the generalizations of the theorem to vector measures (and relations to the geometry of Banach spaces) see Vector measure.
|[a1]||E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)|
Radon-Nikodým theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radon-Nikod%C3%BDm_theorem&oldid=15021