Henstock-Kurzweil integration: its relation to topological vector spaces.

*(English)*Zbl 0954.28001
Series in Real Analysis. 7. Singapore: World Scientific. vii, 136 p. (2000).

In the book the space \(P\) of primitives of Kurzweil-Henstock integrable functions is considered. Recall that \(F\) is primitive to \(f\) if \(F(K)= \int_K f dt\) for any non-degenerate closed subinterval \(K\) of a given compact interval \(I= [a,b]\).

The aim of the book is a topological characterization of so-called \(E\)-convergence of a sequence \((F_i)\) of primitives to a primitive \(F\in P\). The convergence can be characterized by two conditions. The first one is the pointwise convergence \(f_i\to f\), where \(F_i\) are primitives of \(f_i\) and \(F\) is a primitive of \(f\). The second condition is a uniformity condition: for all \(i,j\in N\) and a given sequence \((\delta_j)\) of gauges \(\delta_j: I\to (0,\infty)\) there holds \[ \Biggl|\int_I f_i dt- \sum_k f_i(t_k)|J_k|\Biggr|< 2^{-j}, \] whenever \(\{(t_k, J_k); k= 1,\dots, p\}\) is a \(\delta_j\)-fine partition (i.e., \(\bigcup_k J_k= I\), \(J_k\) are non-overlapping, \(t_k\in J_k\) and \(J_k\subset (t_k- \delta_j(t_k), t_k+\delta_j(t_k))\) for any \(k\)).

The main result of the book is the construction of a topology \({\mathcal T}\) on the space \(P\) of all primitives such that \(E\)-convergence of \((F_i)\) to \(F\) implies its convergence in \((P,{\mathcal T})\) and such that \((P,{\mathcal T})\) is complete and \((P,{\mathcal T})\) is a topological vector space. Of course, the corresponding theory has a more general counterpart.

The book is self-contained, in the first chapter basic information are exposed about the Kurzweil integral (with hints of proofs and references). The main result presented here is the characterization of integrable functions and their primitives. The same is realized in Chapter 2, of course, the corresponding gauges \(\delta_i\) are supposed to be Borel measurable. This result is important for the proof (in Chapter 8) of the completeness of the space \((P,{\mathcal T})\).

In Chapter 3 a new kind of convergence in \(P\) is introduced, so-called \(Q\)-convergence, and it is proved that \(E\)-convergence implies \(Q\)-convergence. The concept is generalized in Chapter 4. Instead of the Banach space \(A\supset P\) of all additive and continuous functions which map subintervals of \(I\) to the reals, an arbitrary Banach space \(X\) is considered. Instead of the set of all sequences of gauges an arbitrary set \(D\) is taken and the \(Q\)-convergence is considered with respect to a mapping \(Q\) from \(D\) to the set of all convex balanced closed subsets of \(X\) such that for any \(d_1,d_2\in D\) there exists \(d_3\in D\) such that \(Q(d_1)+ Q(d_2)\subset Q(d_3)\). A sequence \((x_i)\) \(Q\)-converges to \(x\), if there exists \(d\in D\) such that \(x_i\in Q(d)\) for any \(i\) and \(x_i\to x\) in \(X\).

Chapter 7 contains the construction (in the general setting) of a topology \({\mathcal T}\) on \({\mathbf P}= \bigcup\{Q(d); d\in D\}\) such that \(Q\)-convergence implies \({\mathcal T}\)-convergence. It follows that in the special case \({\mathbf P}= P\) of the set \(P\) of primitives \(E\)-convergence implies \({\mathcal T}\)-convergence. Chapter 8 contains the complete discussion about this special but the most important case.

Chapters 5 and 6 are not necessary for the proof of the main theorem. Chapter 5 in the abstract setting is concerned in the case \(D=N\) and Chapter 6 contains some important facts about \(Q\)-convergence in the space \(P\) of primitives.

Some open problems are included in Chapter 9.

Although the results of the book are very interesting and profound, the book can be read successfully without preliminary knowledge. It is written with a great didactical mastery, clearly and precisely.

The whole material including Appendix on topological vector space, Index and References has only 136 pages. It can be recommended not only for specialists on integration theory, but also for a large scale of readers, mainly for postgraduate students.

The aim of the book is a topological characterization of so-called \(E\)-convergence of a sequence \((F_i)\) of primitives to a primitive \(F\in P\). The convergence can be characterized by two conditions. The first one is the pointwise convergence \(f_i\to f\), where \(F_i\) are primitives of \(f_i\) and \(F\) is a primitive of \(f\). The second condition is a uniformity condition: for all \(i,j\in N\) and a given sequence \((\delta_j)\) of gauges \(\delta_j: I\to (0,\infty)\) there holds \[ \Biggl|\int_I f_i dt- \sum_k f_i(t_k)|J_k|\Biggr|< 2^{-j}, \] whenever \(\{(t_k, J_k); k= 1,\dots, p\}\) is a \(\delta_j\)-fine partition (i.e., \(\bigcup_k J_k= I\), \(J_k\) are non-overlapping, \(t_k\in J_k\) and \(J_k\subset (t_k- \delta_j(t_k), t_k+\delta_j(t_k))\) for any \(k\)).

The main result of the book is the construction of a topology \({\mathcal T}\) on the space \(P\) of all primitives such that \(E\)-convergence of \((F_i)\) to \(F\) implies its convergence in \((P,{\mathcal T})\) and such that \((P,{\mathcal T})\) is complete and \((P,{\mathcal T})\) is a topological vector space. Of course, the corresponding theory has a more general counterpart.

The book is self-contained, in the first chapter basic information are exposed about the Kurzweil integral (with hints of proofs and references). The main result presented here is the characterization of integrable functions and their primitives. The same is realized in Chapter 2, of course, the corresponding gauges \(\delta_i\) are supposed to be Borel measurable. This result is important for the proof (in Chapter 8) of the completeness of the space \((P,{\mathcal T})\).

In Chapter 3 a new kind of convergence in \(P\) is introduced, so-called \(Q\)-convergence, and it is proved that \(E\)-convergence implies \(Q\)-convergence. The concept is generalized in Chapter 4. Instead of the Banach space \(A\supset P\) of all additive and continuous functions which map subintervals of \(I\) to the reals, an arbitrary Banach space \(X\) is considered. Instead of the set of all sequences of gauges an arbitrary set \(D\) is taken and the \(Q\)-convergence is considered with respect to a mapping \(Q\) from \(D\) to the set of all convex balanced closed subsets of \(X\) such that for any \(d_1,d_2\in D\) there exists \(d_3\in D\) such that \(Q(d_1)+ Q(d_2)\subset Q(d_3)\). A sequence \((x_i)\) \(Q\)-converges to \(x\), if there exists \(d\in D\) such that \(x_i\in Q(d)\) for any \(i\) and \(x_i\to x\) in \(X\).

Chapter 7 contains the construction (in the general setting) of a topology \({\mathcal T}\) on \({\mathbf P}= \bigcup\{Q(d); d\in D\}\) such that \(Q\)-convergence implies \({\mathcal T}\)-convergence. It follows that in the special case \({\mathbf P}= P\) of the set \(P\) of primitives \(E\)-convergence implies \({\mathcal T}\)-convergence. Chapter 8 contains the complete discussion about this special but the most important case.

Chapters 5 and 6 are not necessary for the proof of the main theorem. Chapter 5 in the abstract setting is concerned in the case \(D=N\) and Chapter 6 contains some important facts about \(Q\)-convergence in the space \(P\) of primitives.

Some open problems are included in Chapter 9.

Although the results of the book are very interesting and profound, the book can be read successfully without preliminary knowledge. It is written with a great didactical mastery, clearly and precisely.

The whole material including Appendix on topological vector space, Index and References has only 136 pages. It can be recommended not only for specialists on integration theory, but also for a large scale of readers, mainly for postgraduate students.

Reviewer: B.Riečan (Bratislava)

##### MSC:

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

28A25 | Integration with respect to measures and other set functions |

26A39 | Denjoy and Perron integrals, other special integrals |

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

46E15 | Banach spaces of continuous, differentiable or analytic functions |