Editing Radon–Nikodym theorem (section)

==The assumption of σ-finiteness==
The Radon–Nikodym theorem above makes the assumption that the measure ''μ'' with respect to which one computes the rate of change of ''ν'' is [[σ-finite measure|σ-finite]].

===Negative example===
Here is an example when ''μ'' is not σ-finite and the Radon–Nikodym theorem fails to hold.

Consider the [[Borel algebra|Borel σ-algebra]] on the [[real line]]. Let the [[counting measure]], {{mvar|μ}}, of a Borel set {{mvar|A}} be defined as the number of elements of {{mvar|A}} if {{mvar|A}} is finite, and {{math|∞}} otherwise. One can check that {{mvar|μ}} is indeed a measure. It is not {{mvar|σ}}-finite, as not every Borel set is at most a countable union of finite sets. Let {{mvar|ν}} be the usual [[Lebesgue measure]] on this Borel algebra. Then, {{mvar|ν}} is absolutely continuous with respect to {{mvar|μ}}, since for a set {{mvar|A}} one has {{math|1=''μ''(''A'') = 0}} only if {{mvar|A}} is the [[empty set]], and then {{math|''ν''(''A'')}} is also zero.

Assume that the Radon–Nikodym theorem holds, that is, for some measurable function {{math|''f''}} one has
:<math>\nu(A) = \int_A f \,d\mu</math>

for all Borel sets. Taking {{mvar|A}} to be a [[singleton set]], {{math|1=''A'' = {''a''}<nowiki/>}}, and using the above equality, one finds
:<math> 0 = f(a)</math>

for all real numbers {{mvar|a}}. This implies that the function {{math| ''f'' }}, and therefore the Lebesgue measure {{mvar|ν}}, is zero, which is a contradiction.

===Positive result===
Assuming <math>\nu\ll\mu,</math> the Radon–Nikodym theorem also holds if <math>\mu</math> is [[Measure (mathematics)#Localizable measures|localizable]] and <math>\nu</math> is ''accessible with respect to'' <math>\mu</math>,<ref name=BP>{{cite book |last1=Brown |first1=Arlen |last2=Pearcy |first2=Carl |title=Introduction to Operator Theory I: Elements of Functional Analysis |date=1977 |isbn=978-1461299288}}</ref>{{rp|at=p. 189, Exercise 9O}} i.e., <math>\nu(A)=\sup\{\nu(B):B\in{\cal P}(A)\cap\mu^\operatorname{pre}(\R_{\ge0})\}</math> for all <math>A\in\Sigma.</math><ref>{{cite book |last1=Fonseca |first1=Irene |last2=Leoni |first2=Giovanni |title=Modern Methods in the Calculus of Variations: L<sup>p</sup> Spaces |publisher=Springer |isbn=978-0-387-35784-3 |page=68}}</ref>{{rp|at=Theorem 1.111 (Radon–Nikodym, II)}}<ref name=BP/>{{rp|at=p. 190, Exercise 9T(ii)}}