From: "G. A. Edgar"
Subject: Re: Q: what's beyond Bochner integrals?
Date: Mon, 03 May 1999 13:50:26 -0400
Newsgroups: sci.math
Keywords: Pettis integral, Birkhoff integral
In article , Volker W. Elling
wrote:
> I have been reading into Bochner integrals recently (used for integration of
> f:(D,A,m)->X where (D,A,m) is some measure space and X a Banach space
> rather than IR or IC). The assumptions made for
> measurability and integrability seem rather strong to me.
> One thing I don't like about the Bochner integral is the following
> example:
>
> Consider the function g:[0,1]x[0,1]->IR,
> g(x,y):= 1 if x Classical integration:
> h(y) := \int_0^1 g(x,y) dx = y
> Now rewrite the situation:
> X = { f:[0,1]->IR }, |f|_X := sup_{x \in [0,1]} |f(x)|
> If I define
> F:[0,1]->X, F(x)(y):=g(x,y),
> and try to compute the Bochner integral \int_0^1 F(x) dx, I fail because
> F is not almost-separably-valued. I "played dumb" here because I could
> have chosen X := L^1[0,1] instead which is separable. However, I am
> interested in an integral definition that does not require functions to
> be a.s.v. and yields h as the integral value.
This example is from Birkhoff (1935), and is used in one of my
papers below.
>
> ++ Are there generalizations of the Bochner integral? In particular, is the
> "weak integral"
> \int f := g if \phi(g) = \int_D \phi(f(x)) dm(x) for all \phi\in X'
> a useful (and known) definition? In the above example, it produces the
> proper result.
That is related to the "Pettis integral". The actual definition
of the Pettis integral requires something like that to work for
all measurable subsets of D. [Since, unlike the Bochner integral,
existence of your weak integral on D need not imply its existence
on subsets of D.] The Pettis integral exists in more general
circumstances than the Bochner integral; but it lacks some
desirable properties. There are some integrals intermediate
between these, such as the Birkhoff integral. There are some still
more general integrals, where the values need not even be
in X but in the second dual X**.
> ++ What journals/books deal with integration of "Banach-valued" functions?
A lot of this is in Dunford-Schwartz volume 1. A survey by Hildebrandt
in Bull. Amer. Math. Soc. 59 (1953) p. 111, deals with what was
known up to that time. This study has been pretty much "out of
fashion" since the 50's, but that shouldn't stop interested
mathematicians from their investigations of it.
> ++ What is the purpose of 'weak measurability' as defined in Yosida's
> "Functional analysis" ?
There is a nice theorem of Pettis saying that if f is weakly measurable
and almost separably valued, then it is Bochner measurable.
My work on this area (from 20 years ago) is in Indiana Univ. Math. J.
26 (1977) p. 663; and 28 (1979) p. 630. Talagrand's book
takes some of it further: Memoirs of the Amer. Math. Soc. no. 307.
--
Gerald A. Edgar edgar@math.ohio-state.edu
Department of Mathematics telephone: 614-292-0395 (Office)
The Ohio State University 614-292-4975 (Math. Dept.)
Columbus, OH 43210 614-292-1479 (Dept. Fax)