The Banach integral is elegant in its definition, and I am intrigued as to why it is so rarely seen. Is it in practice difficult to calculate from the definition? And are there any other problems with it? I would also be interested to see examples of functions that are Banach-integrable, but not Lebesgue-integrable, and functions (if any) whose Banach and Lebesgue integrals over some interval are defined but different. Added in response to comments: The Banach integral is defined in The Encyclopaedic Dictionary of Mathematics. Like the familiar Riemann integral, it's an integral for real functions over a real interval, and not applied to general Banach spaces (except perhaps in some generalization that I know nothing about).
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asked May 20, 2012 at 19:54
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$\begingroup$ Do you mean this? projecteuclid.org/… "S. Banach has introduced an integral which has no convergence properties..." $\endgroup$– Gerald EdgarCommented May 20, 2012 at 21:17
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$\begingroup$ @Gerald: I did look at this paper in my prior search, but it doesn't seem to answer my question---at least in any way that I can understand. $\endgroup$– John BentinCommented May 20, 2012 at 21:47
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1$\begingroup$ I think Gerald's question is "is this what you mean by Banach integral?" This does not seem to be a completely standard term (in that it has no Wikipedia article...). $\endgroup$– Qiaochu YuanCommented May 21, 2012 at 1:49
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2$\begingroup$ I have worked a bit in Banach space theory but have not previously seen the term "Banach integral". $\endgroup$– Bill JohnsonCommented May 21, 2012 at 4:18
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2 Answers
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One can find some information in the German language book Reelle Zahlen by Oliver Deiser. Banach apparently introduced his integral in the paper Sur le problème de la mesure, Fund. Math. 4, 1923. It was apparently introduced to show that a translation invariant and finitely additive extension of Lebesgue measure on all sets of real numbers exist.
The Banach integral of a Riemann-integrable function coincides with the Riemann integral. There is a function whose Banach integral is $0$ that is not Lebesgue integrable. Also there are Lebesgue integrable functions whose Banach integral differs from The Lebesgue integral. The Banach integral is linear, but not "countable additive". The last fact explains that one cannot do all the nice limiting arguments one is used to when working with the Lebesgue integral.
answered May 21, 2012 at 12:45
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I, also, had never heard of the Banach integral. But here are my guesses for your answers:
Is it in practice difficult to calculate from the definition?
Yes, since the definition is an application of the Hahn-Banach theorem, it is more of a theoretical existence than a construction. I assume the functional has many different Hahn-Banach extensions, not just one.
And are there any other problems with it?
Maybe one would be what I quoted in my comment: "...has no convergence properties..."
answered May 21, 2012 at 12:39