Editing Radon–Nikodym theorem (section)

===For signed and complex measures===
If {{mvar|ν}} is a {{mvar|σ}}-finite signed measure, then it can be Hahn–Jordan decomposed as {{math|1=''ν'' = ''ν''<sup>+</sup> − ''ν''<sup>−</sup>}} where one of the measures is finite. Applying the previous result to those two measures, one obtains two functions, {{math|''g'', ''h'' : ''X'' → [0, ∞)}}, satisfying the Radon–Nikodym theorem for {{math|''ν''<sup>+</sup>}} and {{math|''ν''<sup>−</sup>}} respectively, at least one of which is {{mvar|μ}}-integrable (i.e., its integral with respect to {{mvar|μ}} is finite). It is clear then that {{math|1=''f'' = ''g'' − ''h''}} satisfies the required properties, including uniqueness, since both {{mvar|g}} and {{mvar|h}} are unique up to {{mvar|μ}}-almost everywhere equality.

If {{mvar|ν}} is a [[complex measure]], it can be decomposed as {{math|1=''ν'' = ''ν''<sub>1</sub> + ''iν''<sub>2</sub>}}, where both {{math|''ν''<sub>1</sub>}} and {{math|''ν''<sub>2</sub>}} are finite-valued signed measures. Applying the above argument, one obtains two functions, {{math|''g'', ''h'' : ''X'' → [0, ∞)}}, satisfying the required properties for {{math|''ν''<sub>1</sub>}} and {{math|''ν''<sub>2</sub>}}, respectively. Clearly, {{math|1=''f'' = ''g'' + ''ih''}} is the required function.