Uniform absolute continuity

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In mathematical analysis, a collection <math>\mathcal{F}</math> of real-valued and integrable functions is uniformly absolutely continuous, if for every

<math>\epsilon > 0</math>

there exists

<math> \delta>0 </math>

such that for any measurable set <math>E</math>, <math>\mu(E)<\delta</math> implies

<math> \int_E |f| d\mu < \epsilon </math>

for all <math> f\in \mathcal{F} </math>.

See also

• Absolute continuity

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 0070092230