## The generalized McShane integral.(English)Zbl 0810.28006

Developing 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 $$\langle \int \phi_ n\rangle_{n\in \mathbb{N}}$$.

### MSC:

 28B05 Vector-valued set functions, measures and integrals 46G10 Vector-valued measures and integration

### Citations:

Zbl 0551.28001; Zbl 0714.28008; Zbl 0790.28004