1. Introduction ................................................. 5
1.1. Motivation .............................................. 5
1.2. Recent results on Hartman sets, sequences and
functions ............................................... 6
1.3. Content of the paper .................................... 8
2. Measure-theoretic and topological preliminaries .............. 9
2.1. Set algebras  and -functions .......................... 9
2.2. Finitely additive measures and means ................... 11
2.3. Integration on compact spaces .......................... 14
2.4. Compactifications and continuity ....................... 17
2.5. The Stone-Cech compactification βX ..................... 21
2.6. Compactifications, measures, means and Riemann
integral ............................................... 23
2.7. The set of all means ................................... 25
3. Invariance under transformations and operations ............. 27
3.1. Invariant means for a single transformation ............ 27
3.2. Applications ........................................... 29
3.2.1. Finite X ........................................ 29
3.2.2. X = Z, T:x → x+1 ................................ 30
3.2.3. X compact, А = C(X), T continuous ............... 30
3.2.4. Shift spaces and symbolic dynamics .............. 32
3.2.5. The free group F(x, y) .......................... 33
3.3. Compactifications for transformations and actions ...... 33
3.4. Separate and joint continuity of operations ............ 35
3.5. Compactifications for operations ....................... 38
3.6. Invariance on groups and semigroups .................... 39
3.6.1. The action of a semigroup by translations ....... 39
3.6.2. Means ........................................... 40
3.6.3. Measures ........................................ 42
3.6.4. Amenability ..................................... 43
4. Hartman measurability ....................................... 44
4.1. Definition of Hartman functions ........................ 44
4.2. Definition of weak Hartman functions ................... 46
4.3. Compactifications of LCA groups ........................ 48
4.4. Realizability on LCA groups ............................ 49
4.4.1. Preparation ..................................... 49
4.4.2. Estimate from above ............................. 50
4.4.3. Estimate from below ............................. 53
5. Classes of Hartman functions ................................ 54
5.1. Generalized jump discontinuities ....................... 54
5.2. Hartman functions that are weakly almost periodic ...... 57
5.3. Hartman functions without generalized jumps ............ 59
5.4. Hartman functions with small support ................... 60
5.5. Hartman functions on Z ................................. 64
5.5.1. Fourier-Stieltjes transformation ................ 65
5.5.2. Example ......................................... 66
6. Summary ..................................................... 69
References ..................................................... 70
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