Editing Absolute continuity (section)

==Relation between the two notions of absolute continuity==
A finite measure ''μ'' on [[Borel set|Borel subsets]] of the real line is absolutely continuous with respect to [[Lebesgue measure]] if and only if the point function:
:<math>F(x)=\mu((-\infty,x])</math>
is an absolutely continuous real function. More generally, a function is locally (meaning on every bounded interval) absolutely continuous if and only if its [[distributional derivative]] is a measure that is absolutely continuous with respect to the Lebesgue measure.

If absolute continuity holds then the Radon–Nikodym derivative of ''μ'' is equal almost everywhere to the derivative of ''F''.<ref>{{harvnb|Royden|1988|loc=Problem 12.17(b) on page 303}}.</ref>

More generally, the measure ''μ'' is assumed to be locally finite (rather than finite) and ''F''(''x'') is defined as ''μ''((0,''x'']) for {{nowrap|''x'' > 0}}, 0 for {{nowrap|1=''x'' = 0}}, and −''μ''((''x'',0]) for {{nowrap|''x'' < 0}}. In this case ''μ'' is the [[Lebesgue–Stieltjes integration|Lebesgue–Stieltjes measure]] generated by ''F''.<ref>{{harvnb|Athreya|Lahiri|2006|loc=Sect. 1.3.2, page 26}}.</ref> The relation between the two notions of absolute continuity still holds.<ref>{{harvnb|Nielsen|1997|loc=Proposition 15.7 on page 252}}; {{harvnb|Athreya|Lahiri|2006|loc=Theorem 4.4.3 on page 131}}; {{harvnb|Royden|1988|loc=Problem 12.17(a) on page 303}}.</ref>