Jan Mikusinski, "The Bochner Integral"
Academic Press | 1978 | ISBN: 0124958508 | 233 pages | PDF | 3 MB
In the 1930's a great effort was made to develop the basic theory of Banach
space valued functions of a real variable. The pioneers in this study (Bochner,
Dunford, Gelfand, Pettis, Phillips and Richart) developed a number of
integrals of varying strengths for a multitude of purposes: oftentimes, the
representation of operators on concrete spaces was the object; quite as often
a desire just to understand the abstract process of integration was sufficient
motivation. Of the integrals developed, one integral, the Bochner integral,
emerged as the strongest and, to-date, it is the Bochner integral that has been
the most useful.
Curiously, the Bochner integral is the easiest of the vector integrals from
yesteryear to develop and the one with the most transparent structure.
Indeed, most of the usual results valid for the Lebesgue integral easily adapt
to the Bochner setting. One notable exception: the fundamental theorem of
calculus for absolutely continuous functions defined on [0, 1]. It is simply not
the case that an absolutely continuous vector-valued function defined on
[0, 1] need be the indefinite Bochner integral of its derivative-at least not
unless the vector values are suitable chosen. This pathology is not all together
a bad thing. The study of the class of Banach spaces for which the fundamental
theorem remains valid has kept a number of mathematicians busy and
off the streets for the past five years at least. This class of spaces (whose
members answer to the name "Radon-Nikodym") has come to play an
important role in modern Banach space theory especially as it interacts (and
it does so quite nicely) with probability theory, harmonic analysis and the
infinite dimensional topology. Any book purporting to be about the Bochner
integral should at the very least enter into a careful discussion of why the
Bochner integral does not mimic the Lebesgue integral with regards to the
fundamental theorem of calculus. Included in such a presentation ought to be
the classical results of a positive nature regarding the differentiation of
vector-valued absolutely continuous functions. Many other things might be
equally important for application sake, but before we get carried away with
this line of thought let it be said once and for all that the book under review
(no I've not forgotten it!) is not about the Bochner integral! I know the title
says it is and many of its results pertain to the Bochner integral but the
Bochner integral is not the real subject of the book.
To paraphrase Professor Mikusinski slightly, the purpose of the book is to
give an approach to the theory of the Lebesgue integral which would be "as
intelligible and lucid as possible-understandable to students in their first
undergraduate courses."
The aim is laudable and, taken with a touch of reason, it is achieved. The
inclusion of the Bochner integral is due to the fact that Professor Mikusinski's
approach to the Lebesgue integral "extends automatically to the Bochner
integral (by replacing real coefficients of series by elements of a Banach
space)." To be sure, the enthusiasm expressed in the above paragraph of
making the material available to first year undergraduates is excessive but
with that as a goal, Professor Mikusinski attains an interesting and readable
introduction to the Lebesgue integral. The book under review suffers from
several glaring omissions. Most notable absence-exercises! Professor Mikusinski's
approach to the Lebesgue integral is not standard (though not as
novel as one might conclude from the text) and, therefore, exercises might
well serve to buoy the confidence of the reader. Standard topics such as the
Lebesgue spaces and Fourier series are not to be found in this book. On the
other hand, Professor Mikusinski's treatment of "Changes of Variables" is
noteworthy. Also, the fact that he does present the Bochner integral with so
little extra work might be considered a plus for the book. Since so little extra
work is needed, the failure to discuss the validity of a general fundamental
theorem is an "opportunity lost."
In summary, the title of the book is misleading but the contents
worthwhile. It might be that An approach to Lebesgue integration or even
Lebesgue integration made simple would be appropriate as a title of the book
and descriptive of the aims of the author. The Bochner Integral seems
inappropriate in each role.
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