Measurable fields and direct integrals — lifting and differentiation
Two complete lessons: the lifting theorem and Besicovitch covering and differentiation, with five solved exercises.
This edition contains 2 complete selected lessons; it is not the complete parent course.
Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).
Prerequisites and supporting reading · Download the reader and editable sources
The lifting theorem
- 1. Measure algebras
- 2. Adding one element
- 3. Increasing sequences
- 4. Lower densities of probability spaces
- 5. From lower densities to liftings
- 6. The lifting theorem
- 7. Exercises
- Where this leads
- References
Besicovitch's covering theorem and the differentiation of Radon measures
- 1. Radon measures on \(\mathbb R^r\)
- 2. The covering lemma
- 3. Besicovitch's covering theorem
- 4. Disjoint covers up to a null set
- 5. The differentiation theorem
- 6. Exercises
- Where this leads
- References
Sources and terms
English · original exposition under CC0 1.0 · human sources retain their credit and terms.