Contents
1 Introduction 1
2 Covering relations 3
2.1 Basic language of covering relations 5
2.2 Full and fine covering relations 6
2.3 Covering lemmas 8
3 The variation 11
3.1 Variation of an interval-point function 12
3.2 Differential equivalence 13
3.3 Variational measures 15
3.4 Increasing sets property 17
3.5 Regularity properties 20
3.6 The upper integral . 22
3.7 The variational measure as an integral 25
3.8 The fine variational measure as an integral 28
3.9 Differential equivalence 30
3.10 A density theorem 31
4 Derivates 35
4.1 Definitions of the derivates 35
4.2 Baire class of derivates 36
4.3 Variational estimates 37
4.4 Lipschitz conditions . . , 39
4.5 Exact derivatives 42
5 Absolute continuity and singularity 43
5.1 Basic Definitions 44
5.2 Further properties 46
5.3 Measure properties 48
5.4 A stronger orthogonality relation 50
5.5 Derivation properties and singularity 53
5.6 Characterization of singularity 55
5.7 Derivation properties and absolute continuity 57
5.8 Characterization of absolute continuity 58
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