Radon-Nikodým theorem

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 [1] and O.M. Nikodým [2]. 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 [4]) generalizations of the theorem to the case when the charge takes values in some vector space.

References

[1]	J. Radon, "Ueber lineare Funktionaltransformationen und Funktionalgleichungen" Sitzungsber. Acad. Wiss. Wien , 128 (1919) pp. 1083–1121
[2]	O.M. Nikodým, "Sur une généralisation des intégrales de M. J. Radon" Fund. Math. , 15 (1930) pp. 131–179
[3]	N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958)
[4]	J. Diestel, J.J. Uhl jr., "Vector measures" , Math. Surveys , 15 , Amer. Math. Soc. (1977)

Comments

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.

References

[a1]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965)