Editing Radon–Nikodym theorem (section)

===For {{mvar|σ}}-finite positive measures===
If {{mvar|μ}} and {{mvar|ν}} are {{mvar|σ}}-finite, then {{mvar|X}} can be written as the union of a sequence {{math|{''B<sub>n</sub>''}<sub>''n''</sub>}} of [[disjoint sets]] in {{math|Σ}}, each of which has finite measure under both {{mvar|μ}} and {{mvar|ν}}. For each {{mvar|n}}, by the finite case, there is a {{math|Σ}}-measurable function {{math| ''f<sub>n</sub>''  : ''B<sub>n</sub>'' → [0, ∞)}} such that
:<math>\nu_n(A) = \int_A f_n\,d\mu</math>

for each {{math|Σ}}-measurable subset {{mvar|A}} of {{math|''B<sub>n</sub>''}}. The sum <math display="inline">\left(\sum_n f_n 1_{B_n}\right) := f</math> of those functions is then the required function such that <math display="inline">\nu(A) = \int_A f \, d\mu</math>. 

As for the uniqueness, since each of the {{math|''f<sub>n</sub>''}} is {{mvar|μ}}-almost everywhere unique, so is {{math|''f''}}.