Contact 
This email address is being protected from spambots. You need JavaScript enabled to view it.



Real Functions  Current Topics
Lecture Notes in Mathematics 1603, Springer Verlag 1995
The preface
The Lebesgue integral, introduced at the beginning of our century has been a turning point in the development of the Mathematical Analysis in general and for the evolution of the notion of integral in particular. Most books devoted in our century to the theory of the integral are dominated by the ideas having their source in Lebesgue's theory. One of these ideas is the property of absolute convergence of the Lebesgue integral (f is integrable if and only if f  is integrable), very convenient in most fields of Functional Analysis.
However it is wellknown, the theory of trigonometric series and other lines of thinking in Mathematical Analysis lead to the introduction of some generalizations of the Lebesgue integral that are no longer absolutely convergent. The first monograph giving a clear account of these nonabsolute integrals (for instance, Perron's integral and Denjoy's integrals) was due to Stanislaw Saks (Theory of the Integral, 2nd. rev. ed. Monografie Matematyczne, PWN Warsaw 1937). But this phenomenon remained an isolated one; most subsequent books devoted to the theory of the integral ignored the nonabsolute integrals, despite the fact that the literature devoted to them became richer and richer. The aim of the present monograph is just to fill this gap.
We don't ignore the existence of several books concerned in the last decade, with the KurzweilHenstock integral, one of the most important nonabsolute integral; but as it is well known, this integral is equivalent to the Perron integral, while our aim is to go further and to cover the next steps in the literature devoted to nonabsolute integrals. In this respect, an outstanding result is the Foran integral (1975), which is a classical generalization of the Denjoy integral in the wide sense. In 1986, Iseki gave another generalization of the Denjoy integral, and called it the sparse integral. However neither of the two generalizations is contained one in the other. Our purpose is to give a descriptive definition of an integral which contains both, Foran's integral and Iseki's integral. In order to do this, and also to classify the so many existing integrals, it seems natural to perform a very deep study on the large number of classes of real functions, which have been introduced in this context (many of them by the author himself).
Since there are necessary a lot of examples and counterexamples (66) to illustrate the properties of various classes of functions, we considered that it would be useful to gather them in a separate chapter, at the end of the book.
To show the uniqueness of the integrations we need several monotonicity theorems, which are studied in Chapter IV. But this part of the book is also interesting by itself, because we give there strong generalizations of many monotonicity theorems, studied in the books of Saks (Theory of the Integral, 2nd. rev. ed. Monografie Matematyczne, PWN Warsaw 1937), Bruckner (Differentiation of Real Functions, Lect. Notes in Math., 659, SpringerVerlag 1978) and Thomson (Real functions, Lecture Notes in Mathematics, 1170, SpringerVerlag 1985) (often Lusin's condition (N) is replaced by Foran's condition (M)).
Thomson introduced in 1982 the notion of local systems, trying to unify the theory of the derivation and integration processes, and to indicate the close connection between them (Derivation bases on the real line, I and II. Real Analysis Exchange, 8 (1982/3), 67208, 280442; Real Functions, Lecture Notes in Mathematics, 1170, SpringerVerlag 1985). Since then many authors used local systems in the study of continuity, derivability, monotonicity and integrability. Beside monotonicity, we use local systems to investigate complete Riemann integrals (KurzweilHenstock integrals). We give Perron and descriptive type definitions for these integrals, and show that they are contained in Denjoy type integrals (in the wide sense).
Chapter V we extend the classical result of Nina Bary, that a continuous function on an interval is the sum of three superpositions of absolutely continuous functions (Mémoire sur la représentation finie des fonctions continues, Math. Ann., 103 (1930), 185248 and 598653). We also show the close relationship between Nina Bary's wrinkled functions and Foran's condition (M).
The present monograph can be used as a textbook of a special course of Mathematical Analysis, devoted to graduate and postgraduate students in Mathematics. By its systematic nature and by the great attention given to examples, this monograph should be appropriate for all students who want to deepen their knowledge and understanding of one of the most important chapters of Mathematical Analysis.
In writing this monograph, we have benefited from discussions with several persons. Let us mention first our ten month stay at the Michigan State University, East Lansing, 19911992, giving us the opportunity of frequent talks with Professor J.Marik and Professor Clifford E.Weil. During the same stay we met Professor Paul Humke who was the first to suggest us the idea of writing this monograph, after an attempt to write a survey article on a similar topic.
We received a significant help from the anonymous reviewers of this monograph, for editorial purposes.
Professor Solomon Marcus, who guided us in our first steps in Real Analysis as well as in our Ph.D. thesis, helped us in many respects in improving this monograph.
The Springer Publishing House gave us an important support in improving the final presentation.
To all these persons, to Springer Publishing House and last but not least to our wife Gabriela Ene, who took care of technical aspects related to English and printing, we express our warmest thanks.
Vasile Ene
The contents
1 

Preliminaries 
1 

1.1 
Notations 
1 

1.2 
The δdecomposition of a set 
2 

1.3 
Notions related to Hausdorff measures. Conditions f.l. and σ.f.l. 
2 

1.4 
Oscillations 
4 

1.5 
Borel sets F_{σ}, G_{δ} Borel functions; Analytic sets 
8 

1.6 
Densities; First category sets 
9 

1.7 
The Baire Category Theorem; Romanovski's Lemma 
10 

1.8 
Vitali's Covering Theorem 
11 

1.9 
The generalized properties PG, [PG], P_{1};P_{2}G 
11 

1.10 
Extreme derivatives 
13 

1.11 
Approximate continuity and derivability 
14 

1.12 
Sharp derivatives D^{#}F 
15 

1.13 
Local systems; examples 
16 

1.14 
open sets 
19 

1.15 
Semicontinuity;semicontinuity 
21 
2 

Classes of functions 
25 

2.1 
Darboux conditions 
25 

2.2 
Baire conditions 
27 

2.3 
Conditions 
31 

2.4 
Conditions internal, internal*, 
33 

2.5 
Conditions 
36 

2.6 
Conditions and^{ } local systems 
39 

2.7 
Conditions VB, VB, VBG, 
41 

2.8 
Conditions VB*, VB*, VB*G 
44 

2.9 
Conditions monotone* and VB* 
47 

2.10 
Conditions VB*, VB*G and, lower internal, internal 
50 

2.11 
Conditions AC, ACG 
51 

2.12 
Conditions AC*, AC*, AC*G, AC*G 
54 

2.13 
Conditions L, L, LG, LG 
58 

2.14 
Summability and conditions VB and AC 
60 

2.15 
Differentiability and conditions VBG, VB*G 
62 

2.16 
A fundamental lemma for monotonicity 
65 

2.17 
Krzyzewski's lemma and Foran's lemma 
70 

2.18 
Conditions (N), T_{1}, T_{2}, (S), (+), () 
71 

2.19 
Conditions wS, wN 
78 

2.20 
Condition (N) 
78 

2.21 
Conditions N^{∞} , N^{+∞}, N^{∞} 
79 

2.22 
Conditions M*,M* 
82 

2.23 
Conditions (M), M , N_{g}^{∞} , N_{g}^{+∞} 
84 

2.24 
Derivation bases 
87 

2.25 
Conditions AC_{D#}, AC_{Do}, AC_{D} 
88 

2.26 
Conditions Y_{D#}, Y_{Do}, Y_{D} 
89 

2.27 
Characterizations of, AC and AC 
90 

2.28 
Conditions AC_{n}, AC_{ω} , AC_{∞} , 
93 

2.29 
Conditions VB_{n}, VB_{ω} , VB_{∞} , 
96 

2.30 
Variations V_{n}, V_{ω} , V_{∞} and the Banach Indicatrix 
100 

2.31 
Conditions S_{o}, wS_{o} and AC_{∞} , VB_{∞} , (N) 
103 

2.32 
Conditions L_{n}, L_{ω} , L_{∞} , 
104 

2.33 
Conditions ΛZ, f.l., σ.f.l. 
108 

2.34 
Conditions SAC_{n}, SAC_{ω}, SAC_{∞},, 
111 

2.35 
Conditions SVB_{n}, SVB_{ω}, SVB_{∞},, 
114 

2.36 
Conditions DW_{n}, DW_{ω} , DW_{∞} , DW* 
116 

2.37 
Conditions E_{n}, E_{ω}, E_{∞},, 
118 

2.38 
Conditions 
121 
3 

Finite representations for continuous functions 
127 

3.1 
quasiderivable ⊆ AC*; DW*G + AC*; DW*G and approximately quasiderivable ⊆ AC; DW_{1}G+AC; DW_{1}G 
127 

3.2 
⊆ DW_{1}+DW_{1} on a perfect nowhere dense set 
130 

3.3 
Wrinkled functions (W) and condition (M) 
131 

3.4 
= quasiderivable + quasiderivable 
134 

3.5 
= AC*; DW_{1}G+AC*; DW_{1}G+AC*; DW_{1}G 
135 
4 

Monotonicity 
141 

4.1 
Monotonicity and conditions (), VB_{ω}G, 
141 

4.2 
Monotonicity and conditions (M), uCM, AC, 
142 

4.3 
Monotonicity and conditions N^{∞}; N^{∞} 
145 

4.4 
Local monotonicity 
149 

4.5 
derivatives and the Mean Value Theorem 
149 

4.6 
Relative monotonicity 
151 

4.7 
An application of Corollary 4.4.1 
151 

4.8 
A general monotonicity theorem 
152 

4.9 
Monotonicity in terms of extreme derivatives 
158 
5 

Integrals 
161 

5.1 
Descriptive and Perron type definitions for the Lebesgue integral 
161 

5.2 
Ward type definitions for the Lebesgue integral 
166 

5.3 
Henstock variational definitions for the Lebesgue integral 
167 

5.4 
Riemann type definitions for the Lebesgue integral (The McShane integral) 
169 

5.5 
Theorems of Marcinkiewicz type for the Lebesgue integral 
172 

5.6 
Bounded Riemann^{#} sums; locally small Riemann^{#} sums 
173 

5.7 
Descriptive and Perron type definitions for theintegral 
174 

5.8 
An improvement of the Hake Theorem 
180 

5.9 
An improvement of the LoomanAlexandroff Theorem. The equivalence of theintegral and the (P_{j,k})integral 
184 

5.10 
Ward type definitions for theintegral 
186 

5.11 
Henstock variational definitions for theintegral 
187 

5.12 
The KurzweilHenstock integral 
188 

5.13 
Cauchy and Harnak extensions of theintegral 
189 

5.14 
A theorem of Marcinkiewicz type for theintegral 
190 

5.15 
Bounded Riemann sums and locally small Riemann sums 
192 

5.16 
Riemann type integrals and local systems 
193 

5.17 
The <LPG> and <LDG> integrals 
197 

5.18 
The chain rule for the derivative of a composite function 
200 

5.19 
The chain rule for the approximate derivative of a composite function 
202 

5.20 
Change of variable formula for the Lebesgue integral 
204 

5.21 
Change of variable formula for the Denjoy*integral 
205 

5.22 
Change of variable formula for the <LDG>integral 
206 

5.23 
Integrals of Foran type 
207 

5.24 
Integrals which extend both, Foran's integral and Iseki's integral 
210 
6 

Examples 
213 

6.1 
The Cantor ternary set, a perfect nowhere dense set 
213 

6.2 
The Cantor ternary function φ 
214 

6.3 
A real bounded S_{o}^{+} closed set which is not of F_{σ}type 
214 

6.4 
An S_{o}^{+} lower semicontinuous function which is not 
215 

6.5 
A function 
215 

6.6 
A function 
215 

6.7 
A function 
216 

6.8 
A function F ∈ uCM; F ∉ lCM 
216 

6.9 
A function concerning conditions, CM, sCM, lower internal 
217 

6.10 
A function concerning conditions , internal, [VBG], (), T_{1}, T_{2} (Bruckner) 
217 

6.11 
A function concerning conditions, lower internal, internal, internal* (Dirichlet) 


6.12 
A function concerning conditions:, lower internal, internal*, VB, VB*G, N^{∞} 


6.13 
A function 


6.14 
A function , F ∈ lower internal, 
220 

6.15 
A function F ∈ sCM, F ∉ internal* 
220 

6.16 
A function F ∈ AC*G \ AC, F ∈ sCM \ internal* 
220 

6.17 
A function F ∈ (D.C.),, F ∉ m_{2}, 
221 

6.18 
A function F ∈ (+) ∩ (); 
221 

6.19 
A function , G'_{ap}(x) exists n.e., G'_{ap}(x) ≥ 0 a.e. (Preiss) 
222 

6.20 
A function H'_{ap}(x) exists on (0,1) (Preiss) 
222 

6.21 
A function F(x) = 0 a.e., F is not identically zero (Croft) 
223 

6.22 
A function , F ∈ [VBG], F ∉ VB*G, (Bruckner) 
223 

6.23 
A function F ∈ AC*, F ∉ VB* 
224 

6.24 
A function F ∈, F ∈ T_{1}, F ∈ VBG, F ∉ VB*G 
224 

6.25 
A function F ∈ [bAC*G] ∩ VB*G N^{∞}, F ∉ lower internal 
225 

6.26 
A function F ∈ C ∩ (S) ∩ LG, F ∉ AC*G, F'(x) does not exist on a set of positive measure, F(x)+x ∈ LG, F(x)+x ∉ T_{1} 
225 

6.27 
A function F ∈ (S) ∩ C such that the sum of F and any linear nonconstant function does not satisfy (N) (Mazurkievicz) 
226 

6.28 
A function F ∈ (M), F ∉ T_{2} 
227 

6.29 
Functions concerning conditions (M), AC, T_{1}, T_{2}, (S), (N), L, L_{2}G, VBG,, quasiderivable 
229 

6.30 
A function G ∈ N^{∞} , F ∉ (M), F ∉ (+) 
237 

6.31 
Functions concerning conditions (S), (N), (M), T1, T_{2}, ACG, AC_{n}, SAC_{n}, VB_{2}, VBG, SVB, 
238 

6.32 
A function F ∈ lower semicontinuous, F ∈ AC_{2}, F ∉ AC 
244 

6.33 
A function F_{n} ∈ L_{n+1} on a perfect set, F_{n} ∈ VB_{n} on no portion of this set, F_{n} ∈ L_{n+1}G, F_{n} ∉ AC_{n}G on [0,1] 
245 

6.34 
Functions F ∈ L_2G, G_{s} ∈ (N), G'_{s} = F' a.e., G_{s}F is not identically zero, F ∉ SACG 
247 

6.35 
A function F ∈ L_{2}, F ∉ T_{2}, F ∉ 
250 

6.36 
A function F ∈ VB_{2} on C, V_{2}(F;C) \≤ 1 
252 

6.37 
A function F_{p} ∈ L_{2p}, F_{p} ∉ AC_{2p1}, F_{p} ∈ VB_{2} on C, V_{2}(F_{p};C) ≤ 1 
252 

6.38 
A function G ∈ VB_{2}, G ∉ AC_{n} on C, G ∈ on [0,1] 
254 

6.39 
A function F_{1} ∈ VB_{2} on C, V_{2}(F_{1};[0,x] ∩ C) = φ(x) (G. Ene) 
254 

6.40 
A function F_{q} ∈ (N) on [0,1], F_{q} ∉ VB_{n} on C, F_{q} ∈ VB_{ω} on C (G. Ene) 
255 

6.41 
A function G_{1} ∈ L_{2}G, G_{1} ∈, G_{1} ∉ SVBG, G_{1} ∉ SACG, (G_{1})'_{ap} does not exist on a set of positive measure 
256 

6.42 
A function F ∈ SACG, F ∉, F ∉ ACG 
258 

6.43 
A function F ∈ DW_{1}, F ∉ DW* 
261 

6.44 
A function F ∈ AC*;DW_{1}G, F ∉ AC*;DW*G 
261 

6.45 
Functions F_{1},F_{2} ∈ C ∩ AC*;DW*G, F_{1}, F_{2} are derivable a.e., F'_{1} = F'_{2} a.e., F_{1} and F_{2} do not differ by a constant 
262 

6.46 
Functions F_{1},F_{2} ∈ C ∩ AC*;SW_1G, F_{1},F_{2} are approximately derivable a.e., F_{1}+F_{2} ∉ quasiderivable 
262 

6.47 
Functions F_{1},F_{2} ∈, F_{1},F_{2} ∉, F_{1}+F_{2} ∉ 
264 

6.48 
A function G_{n} ∈ E_{n+1}, G_{n} ∉ E_{n}, G_{n} ∈ L_{n2 + 2n+1}, G_{n} ∉ VB_{n2+2n} 
265 

6.49 
Functions concerning conditions L,, E_{1}G 
268 

6.50 
A function F ∈, F ∉ 
270 

6.51 
A function F ∈ (N), F ∉ ΛZ (Foran) 
271 

6.52 
A function F ∈ AC º ΛZ, F ∉ ΛZ (Foran) 
272 

6.53 
A function H ∈ AC+ΛZ, H ∉ ΛZ (Foran) 
272 

6.54 
A function G ∈ AC · ΛZ, G ∉ ΛZ (Foran) 
272 

6.55 
A function F_{1} ∈ AC, F_{1} ∉ L_{n}, F_{1} ∉ 
273 

6.56 
Functions F_{1} ∈ AC_{2}G, F_{2} ∈ ΛZ, F_{1}+F_{2} ∉ (M), F_{1}' = F_{2}' a.e. 
273 

6.57 
A function F ∈ ΛZ, F ∉ 
274 

6.58 
Functions F_{1} ∈ (S), F_{1} ∈ AC º σ.f.l. F_{1} ∉ σ.f.l., F_{2} ∈ L, F_{1}+F_{2} ∉ T_{2} 
276 

6.59 
Functions G_{1} ∈ σ.f.l., G_{2} ∈ AC, G_{1}+G_{2} ∉ σ.f.l. 
279 

6.60 
Functions H_{1} ∈ σ.f.l., H_{2} ∈ AC, H_{1} · H_{2} ∉ σ.f.l. 
280 

6.61 
A function F ∈ σ.f.l., F ∈ T_{1}, F ∉, F is nowhere approximately derivable (Foran) 
280 

6.62 
A function G ∈ σ.f.l. ∩ T_{1}, G is nowhere derivable, G'_{ap}(x) = 0 a.e., G ∉ W, G ∈ W* (Foran) 
284 

6.63 
A function F ∈ W on a perfect nowhere dense set of positive measure with each level set perfect, F is nowhere approximately derivable 
287 

6.64 
A function G_{1} ∈ DW_{1} ∩ C, G_{1} is not approximately derivable a.e. on a set of positive measure 
291 

6.65 
A function F ∈ C, F is quasiderivable, F ∉ AC º AC + AC 
291 

6.66 
Examples concerning the chain rule for the approximate derivative of a composite function 
291 


Bibliography 
293 


Index 
305 



