Uniform absolute continuity

In mathematical analysis, a collection $\mathcal{F}$ of real-valued and integrable functions is uniformly absolutely continuous, if for every $\epsilon > 0$, there exists $\delta>0$ such that for any measurable set $E$, $\mu(E)<\delta$ implies

$\int_E \!|f|\, \mathrm{d}\mu < \epsilon$

for all $f\in \mathcal{F}$.

See also

• Absolute continuity
• Uniform integrability

References

• J. J. Benedetto (1976). Real Variable and Integration - section 3.3, p. 89. B. G. Teubner, Stuttgart. ISBN 3-519-02209-5
• C. W. Burrill (1972). Measure, Integration, and Probability - section 9-5, p. 180. McGraw-Hill. ISBN 0-07-009223-0