The generalized McShane integral.

*(English)*Zbl 0810.28006Developing ideas of E. J. McShane [“Unified integration” (1983; Zbl 0551.28001)], R. A. Gordon [Ill. J. Math. 34, No. 3, 557-567 (1990; Zbl 0714.28008)], and some joint work of myself and J. Mendoza [Ill. J. Math. 38, No. 1, 127-147 (1994; Zbl 0790.28004)], I describe a general form of the McShane integral for functions from a topological measure space to a Banach space, as follows. If \((S,{\mathfrak T},\Sigma,\mu)\) is a \(\sigma\)-finite outer regular quasi-Radon measure space, a generalized McShane partition of \(S\) is a sequence \(\langle(E_ n,t_ n)\rangle_{n\in\mathbb{N}}\) where \(\langle E_ n\rangle_{n\in\mathbb{N}}\) is a disjoint cover of a conegligible subset of \(S\) by sets of finite measure and \(\langle t_ n\rangle_{n\in\mathbb{N}}\) is a sequence in \(S\); a gauge on \(S\) is a function \(\Delta: S\to {\mathfrak T}\) such that \(t\in \Delta(t)\) for every \(t\in S\); and a generalized McShane partition \(\langle(E_ n, t_ n)\rangle_{n\in \mathbb{N}}\) is subordinate to a gauge \(\Delta\) if \(E_ n\subseteq \Delta(t_ n)\) for every \(n\). Now, if \(X\) is a Banach space, a function \(\phi: S\to X\) is McShane integrable, with integral \(w\in X\), if for every \(\varepsilon> 0\) there is a gauge \(\Delta\) such that \(\limsup_{n\to\infty}\| w- \sum^ n_{i= 0} \mu E_ i\phi(t_ i)\|\leq \varepsilon\) whenever \(\langle(E_ n,t_ n)\rangle_{n\in \mathbb{N}}\) is a generalized McShane partition subordinate to \(\Delta\).

I show that this integral agrees with Gordon’s formulation when \(S= [0,1]\), and with McShane’s when \(X= \mathbb{R}\) and \(S\) is one of the cases he examines. I show that Bochner integrable functions are McShane integrable and that McShane integrable functions are Pettis integrable, so that, in particular, the McShane integral agrees with the ordinary integral whenever \(X= \mathbb{R}\). I show that if the unit ball of \(X^*\) is \(w^*\)- separable (that is, if \(X\) can be embedded in \(\ell^ \infty\)) then a McShane integrable function whose norm has a finite upper integral is Talagrand integrable; and that if \(X\) itself is separable then a function is McShane integrable iff it is Pettis integrable. If either \(\mu\) is a Radon measure or there is no real-valued-measurable cardinal, then a Pettis integrable function \(\phi\) such that \(\phi^{-1}[G]\in \Sigma\) for every open set \(G\subseteq X\) is McShane integrable.

In the last section I give a group of convergence theorems. If \(\langle\phi_ n\rangle_{n\in \mathbb{N}}\) is a sequence of McShane integrable functions such that \(\phi(t)= \lim_{n\to \infty} \phi_ n(t)\) (for the weak topology on \(X\)) for every \(t\in S\), then \(\phi\) will be McShane integrable, with \(\int\phi= \lim_{n\to\infty} \int \phi_ n\) (for the weak topology) if one of the following conditions is satisfied: (A) \(\lim_{n\to\infty} \int_ E \phi_ n\) exists in \(X\), for the weak topology, for every \(E\in \Sigma\); (B) \(\{f\phi_ n: n\in \mathbb{N}, f\in X^*, \| f\|\leq 1\}\) is uniformly integrable. Under condition (B), if \(\phi(t)\) is the norm-limit of \(\langle\phi_ n(t)\rangle_{n\in \mathbb{N}}\) for every \(t\in S\), then \(\int \phi\) will be the norm-limit of \(\langle \int \phi_ n\rangle_{n\in \mathbb{N}}\).

I show that this integral agrees with Gordon’s formulation when \(S= [0,1]\), and with McShane’s when \(X= \mathbb{R}\) and \(S\) is one of the cases he examines. I show that Bochner integrable functions are McShane integrable and that McShane integrable functions are Pettis integrable, so that, in particular, the McShane integral agrees with the ordinary integral whenever \(X= \mathbb{R}\). I show that if the unit ball of \(X^*\) is \(w^*\)- separable (that is, if \(X\) can be embedded in \(\ell^ \infty\)) then a McShane integrable function whose norm has a finite upper integral is Talagrand integrable; and that if \(X\) itself is separable then a function is McShane integrable iff it is Pettis integrable. If either \(\mu\) is a Radon measure or there is no real-valued-measurable cardinal, then a Pettis integrable function \(\phi\) such that \(\phi^{-1}[G]\in \Sigma\) for every open set \(G\subseteq X\) is McShane integrable.

In the last section I give a group of convergence theorems. If \(\langle\phi_ n\rangle_{n\in \mathbb{N}}\) is a sequence of McShane integrable functions such that \(\phi(t)= \lim_{n\to \infty} \phi_ n(t)\) (for the weak topology on \(X\)) for every \(t\in S\), then \(\phi\) will be McShane integrable, with \(\int\phi= \lim_{n\to\infty} \int \phi_ n\) (for the weak topology) if one of the following conditions is satisfied: (A) \(\lim_{n\to\infty} \int_ E \phi_ n\) exists in \(X\), for the weak topology, for every \(E\in \Sigma\); (B) \(\{f\phi_ n: n\in \mathbb{N}, f\in X^*, \| f\|\leq 1\}\) is uniformly integrable. Under condition (B), if \(\phi(t)\) is the norm-limit of \(\langle\phi_ n(t)\rangle_{n\in \mathbb{N}}\) for every \(t\in S\), then \(\int \phi\) will be the norm-limit of \(\langle \int \phi_ n\rangle_{n\in \mathbb{N}}\).

Reviewer: D.H.Fremlin (Essex)

##### MSC:

28B05 | Vector-valued set functions, measures and integrals |

46G10 | Vector-valued measures and integration |