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| | In [[mathematical analysis]], a collection <math>\mathcal{F}</math> of [[Real number|real]]-valued and [[integrable function|integrable]] functions is '''uniformly absolutely continuous''', if for every |
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| | :<math>\epsilon > 0</math> |
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| | there exists |
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| | :<math> \delta>0 </math> |
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| | such that for any [[measurable set]] <math>E</math>, <math>\mu(E)<\delta</math> implies |
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| | :<math> \int_E |f| d\mu < \epsilon </math> |
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| | for all <math> f\in \mathcal{F} </math>. |
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| | ==See also== |
| | * [[Absolute continuity]] |
| | * [[Uniform integrability]] |
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| | ==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 |
| | |
| | [[Category:Mathematical analysis]] |
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| | {{mathanalysis-stub}} |
In mathematical analysis, a collection 
of real-valued and integrable functions is uniformly absolutely continuous, if for every
there exists
such that for any measurable set 
, 
implies
for all 
.
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 0-07-009223-0
Template:Mathanalysis-stub