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# Real Functions - Current Topics

## 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 well-known, 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 Kurzweil-Henstock 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, Springer-Verlag 1978) and Thomson (Real functions, Lecture Notes in Mathematics, 1170, Springer-Verlag 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), 67-208, 280-442; Real Functions, Lecture Notes in Mathematics, 1170, Springer-Verlag 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 (Kurzweil-Henstock 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), 185-248 and 598-653). 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, 1991-1992, 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 δ-decomposition 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 Fσ, Gδ 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], P1;P2G 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 $\small{\mathcal S}$-open sets 19 1.15 Semicontinuity;$\small{\mathcal S}$-semicontinuity 21 2 Classes of functions 25 2.1 Darboux conditions$\small{\mathcal D}, ~{\mathcal D}_-,~{\mathcal D}_+$ 25 2.2 Baire conditions$\small{\mathcal B}_1, ~\underline{\mathcal B}_1, ~\overline{\mathcal B}_1$ 27 2.3 Conditions$\small{\mathcal C}_i; ~{\mathcal C}_i^*; ~[{\mathcal C}_i G];~ [{\mathcal C}G]$ 31 2.4 Conditions internal, internal*,$\small{\mathcal Z}_i, ~uCM$ 33 2.5 Conditions$\small{\mathcal D}_-{\mathcal B}_1, ~ {\mathcal D}{\mathcal B}_1$ 36 2.6 Conditions$\small {\mathcal B}_1, ~ w{\mathcal B}_1, ~{\mathcal S}_2^-$ and$\small {\mathcal S}_2^+$ - 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$\small{\mathcal D}, ~{\mathcal D}_-, ~[{\mathcal C}G], ~ [{\mathcal C}G]$, 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), T1, T2, (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 , Ng∞ , Ng+∞ 84 2.24 Derivation bases 87 2.25 Conditions ACD#, ACDo, ACD 88 2.26 Conditions YD#, YDo, YD 89 2.27 Characterizations of$\small AC*G \cap {\mathcal C}, ~AC*G \cap {\mathcal C}_i$,  AC  and AC 90 2.28 Conditions ACn, ACω , AC∞ ,$\small{\mathcal F}$ 93 2.29 Conditions VBn, VBω , VB∞ , $\small{\mathcal B}$ 96 2.30 Variations Vn, Vω , V∞ and the Banach Indicatrix 100 2.31 Conditions So, wSo and AC∞ , VB∞ , (N) 103 2.32 Conditions Ln, Lω , L∞ ,$\small{\mathcal L}$ 104 2.33 Conditions ΛZ, f.l., σ.f.l. 108 2.34 Conditions SACn, SACω, SAC∞,,$\small S{\mathcal F}$ 111 2.35 Conditions SVBn, SVBω, SVB∞,,$\small S{\mathcal B}$ 114 2.36 Conditions DWn, DWω , DW∞ , DW* 116 2.37 Conditions En, Eω, E∞,,$\small {\mathcal E}$ 118 2.38 Conditions$\small{\mathcal S}AC, {\mathcal S}ACG, {\mathcal S}VB, {\mathcal S}VBG, {\mathcal S}Y$ 121 3 Finite representations for continuous functions 127 3.1 quasi-derivable ⊆ AC*; DW*G + AC*; DW*G and approximately quasi-derivable ⊆ AC; DW1G+AC; DW1G 127 3.2 $\small{\mathcal C}$ ⊆ DW1+DW1 on a perfect nowhere dense set 130 3.3 Wrinkled functions (W) and condition (M) 131 3.4 $\small{\mathcal C}$ = quasi-derivable + quasi-derivable 134 3.5 $\small{\mathcal C}$ = AC*; DW1G+AC*; DW1G+AC*; DW1G 135 4 Monotonicity 141 4.1 Monotonicity and conditions (-), VBωG, $\small{\mathcal D}_{\_}{\mathcal B}_1$ 141 4.2 Monotonicity and conditions (M), uCM, AC, $\small{\mathcal C}_i, ~{\mathcal C}_i^*{\mathcal D}_i$ 142 4.3 Monotonicity and conditions N-∞; N∞ 145 4.4 Local monotonicity 149 4.5 $\small{\mathcal S}$-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 the$\small {\mathcal D}^*$-integral 174 5.8 An improvement of the Hake Theorem 180 5.9 An improvement of the Looman-Alexandroff Theorem. The equivalence of the$\small {\mathcal D}^*$-integral and the (Pj,k)-integral 184 5.10 Ward type definitions for the$\small {\mathcal D}^*$-integral 186 5.11 Henstock variational definitions for the$\small {\mathcal D}^*$-integral 187 5.12 The Kurzweil-Henstock integral 188 5.13 Cauchy and Harnak extensions of the$\small {\mathcal D}^*$-integral 189 5.14 A theorem of Marcinkiewicz type for the$\small {\mathcal D}^*$-integral 190 5.15 Bounded Riemann sums and locally small Riemann sums 192 5.16 Riemann type integrals and local systems 193 5.17 The and 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 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 So+ closed set which is not of Fσ-type 214 6.4 An So+ lower semicontinuous function which is not$\small\overline{\mathcal B}_1$ 215 6.5 A function$\small F\in {\mathcal C}_i, ~ F\notin {\mathcal C}_i^*$ 215 6.6 A function$\small F \in {\mathcal D}, ~F \in [{\mathcal C}_i^*G], ~ F \notin [{\mathcal C}G]$ 215 6.7 A function $\small F \in {\mathcal D B}_1,~F\notin [{\mathcal C}G]$ 216 6.8 A function F ∈ uCM; F ∉ lCM 216 6.9 A function concerning conditions$\small {\mathcal D}_+, ~{\mathcal D}_-$,  CM, sCM, lower internal 217 6.10 A function concerning conditions $\small {\mathcal D}_-, ~{\mathcal D}$, internal,$\small \underline{\mathcal B}_1,~ \overline{\mathcal B}_1,~{\mathcal B}_1, ~w{\mathcal B}_1,$ [VBG], (-), T1, T2 (Bruckner) 217 6.11 A function concerning conditions$\small\overline{\mathcal B}_1, ~\underline{\mathcal B}_1, ~ {\mathcal D}_-,~ {\mathcal D} +$,  lower internal, internal, internal* (Dirichlet) 6.12 A function concerning conditions:$\small{\mathcal D}, ~{\mathcal D}_-,~ {\mathcal B}_1\,,~{\mathcal C}_i\,,~ {\mathcal C}_i^*$, lower internal, internal*, VB, VB*G, N-∞ 6.13 A function $\small F \in {\mathcal D}, ~F\in \underline{\mathcal B}_1 \setminus \overline{\mathcal B}_1, ~-F \in \overline{\mathcal Z}_i \setminus {\mathcal C}_i,~ -F \in {\mathcal D}_-\overline{\mathcal B}_1 \setminus \overline{\mathcal Z}_i$ 6.14 A function $\small F \in \underline{\mathcal B}_1 \setminus \overline{\mathcal B}_1$, F ∈ lower internal,$\small F \notin {\mathcal D}_-$ 220 6.15 A function F ∈ sCM, F ∉ internal* 220 6.16 A function F ∈ AC*G \ AC, $\small F \in {\mathcal C}^*_i \setminus {\mathcal D},$ F ∈ sCM \ internal* 220 6.17 A function F ∈ (D.C.),$\small F \in{\mathcal B}_1$, F ∉ m2, $\small F \notin {\mathcal D}$ 221 6.18 A function F ∈ (+) ∩ (-); $\small F \notin {\mathcal DB}_1T_2$ 221 6.19 A function $\small G \in {\mathcal D},~ G \notin \underline{\mathcal B}_1, ~ G \notin \overline{\mathcal B}_1$, G'ap(x) exists n.e., G'ap(x) ≥ 0 a.e. (Preiss) 222 6.20 A function $\small H \in {\mathcal D}, ~H \notin \overline{\mathcal B}_1, ~H \notin \underline{\mathcal B}_1,$ H'ap(x) exists on (0,1) (Preiss) 222 6.21 A function $\small F \in {\mathcal DB}_1,$ F(x) = 0 a.e., F is not identically zero (Croft) 223 6.22 A function $\small F \in {\mathcal D}, ~F \in [{\mathcal C}G]$, F ∈ [VBG], F ∉ VB*G, $\small F \notin {\mathcal C}$ (Bruckner) 223 6.23 A function F ∈ AC*, F ∉ VB* 224 6.24 A function F ∈$\small{\mathcal C}$, F ∈ T1, 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 ∉ T1 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 ∉ T2 227 6.29 Functions concerning conditions (M), AC, T1, T2, (S), (N), L, L2G, VBG,$\small S{\mathcal F}$, quasi-derivable 229 6.30 A function G ∈ N∞ , F ∉ (M), F ∉ (+) 237 6.31 Functions concerning conditions (S), (N), (M), T1, T2, ACG, ACn, SACn, VB2, VBG, SVB, $\small {\mathcal F}, ~S{\mathcal F}$ 238 6.32 A function F ∈ lower semicontinuous, F ∈ AC2, F ∉ AC 244 6.33 A function Fn ∈ Ln+1 on a perfect set, Fn ∈ VBn on no portion of this set, Fn ∈ Ln+1G, Fn ∉ ACnG on [0,1] 245 6.34 Functions F ∈ L_2G, Gs ∈ (N), G's = F' a.e., Gs-F is not identically zero, F ∉ SACG 247 6.35 A function F ∈ L2, F ∉ T2, F ∉$\small{\mathcal B}$ 250 6.36 A function F ∈ VB2 on C, V2(F;C) \≤ 1 252 6.37 A function Fp ∈ L2p, Fp ∉ AC2p-1, Fp ∈ VB2 on C, V2(Fp;C) ≤ 1 252 6.38 A function G ∈ VB2, G ∉ ACn on C, G ∈$\small {\mathcal F}$ on [0,1] 254 6.39 A function F1 ∈ VB2 on C, V2(F1;[0,x] ∩ C) = φ(x)  (G. Ene) 254 6.40 A function Fq ∈ (N) on [0,1], Fq ∉ VBn on C, Fq ∈ VBω on C  (G. Ene) 255 6.41 A function G1 ∈ L2G, G1 ∈$\small{\mathcal F}$, G1 ∉ SVBG, G1 ∉ SACG, (G1)'ap does not exist on a set of positive measure 256 6.42 A function F ∈ SACG, F ∉$\small{\mathcal F}$, F ∉ ACG 258 6.43 A function F ∈ DW1, F ∉ DW* 261 6.44 A function F ∈ AC*;DW1G,   F ∉ AC*;DW*G 261 6.45 Functions F1,F2 ∈ C ∩ AC*;DW*G,  F1, F2 are derivable a.e., F'1 = F'2 a.e., F1 and F2 do not differ by a constant 262 6.46 Functions F1,F2 ∈ C ∩ AC*;SW_1G,  F1,F2 are approximately derivable a.e., F1+F2 ∉ quasi-derivable 262 6.47 Functions F1,F2 ∈$\small {\mathcal E}\cap{\mathcal B}$, F1,F2 ∉$\small {\mathcal F}$, F1+F2 ∉$\small {\mathcal E}$ 264 6.48 A function Gn ∈ En+1, Gn ∉ En, Gn ∈ Ln2 + 2n+1, Gn ∉ VBn2+2n 265 6.49 Functions concerning conditions L,$\small \underline{\mathcal F}, \overline{\mathcal F}, VB_2G, {\mathcal B}$, E1G 268 6.50 A function F ∈$\small {\mathcal E} \cap VB_\omega G$,  F ∉$\small{\mathcal B}$ 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 F1 ∈ AC, F1 ∉ Ln, F1 ∉ $\small{\mathcal L}$ 273 6.56 Functions F1 ∈ AC2G, F2 ∈ ΛZ, F1+F2 ∉ (M), F1' = -F2' a.e. 273 6.57 A function F ∈ ΛZ, F ∉$\small [{\mathcal E}]$ 274 6.58 Functions F1 ∈ (S), F1 ∈ AC º σ.f.l. F1 ∉ σ.f.l., F2 ∈ L, F1+F2 ∉ T2 276 6.59 Functions G1 ∈ σ.f.l., G2 ∈ AC, G1+G2 ∉ σ.f.l. 279 6.60 Functions H1 ∈ σ.f.l., H2 ∈ AC, H1 · H2 ∉ σ.f.l. 280 6.61 A function F ∈ σ.f.l., F ∈ T1, F ∉$\small{\mathcal B}$, F is nowhere approximately derivable (Foran) 280 6.62 A function G ∈ σ.f.l. ∩ T1, 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 G1 ∈ DW1 ∩ C, G1 is not approximately derivable a.e. on a set of positive measure 291 6.65 A function F ∈ C, F is quasi-derivable, 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

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