Uniform absolute continuity
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
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