Editing Absolute continuity (section)

===Equivalent definitions===

The following conditions on a real-valued function ''f'' on a compact interval [''a'',''b''] are equivalent:<ref>{{harvnb|Nielsen|1997|loc=Theorem 20.8 on page 354}}; also {{harvnb|Royden|1988|loc=Sect. 5.4, page 110}} and {{harvnb|Athreya|Lahiri|2006|loc=Theorems 4.4.1, 4.4.2 on pages 129,130}}.</ref>

# ''f'' is absolutely continuous;
# ''f'' has a derivative ''f''&nbsp;{{prime}} [[almost everywhere]], the derivative is Lebesgue integrable, and <math display="block"> f(x) = f(a) + \int_a^x f'(t) \, dt </math> for all ''x'' on [''a'',''b''];
# there exists a Lebesgue integrable function ''g'' on [''a'',''b''] such that <math display="block"> f(x) = f(a) + \int_a^x g(t) \, dt </math> for all ''x'' in [''a'',''b''].

If these equivalent conditions are satisfied, then necessarily any function ''g'' as in condition 3. satisfies ''g'' = ''f''&nbsp;{{prime}} almost everywhere.

Equivalence between (1) and (3) is known as the '''fundamental theorem of Lebesgue integral calculus''', due to [[Lebesgue]].<ref>{{harvnb|Athreya|Lahiri|2006|loc=before Theorem 4.4.1 on page 129}}.</ref>

For an equivalent definition in terms of measures see the section [[#Relation between the two notions of absolute continuity|Relation between the two notions of absolute continuity]].