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On the distinguishing features of the Dobrakov integral. (English) Zbl 0883.28011
The paper under review examines the distinguishing features of the Dobrakov integration theory of vector-valued functions with respect to an operator-valued measure which is countably additive in the strong operator topology. This theory was developed in a series of 15 papers that Dobrakov published in Czechoslovak Mathematical Journal between 1970 and 1990. We present briefly the setting of this theory.
Let $$T$$ be a non-empty set, let $$\mathcal P$$ be a $$\delta$$-ring of subsets of $$T$$ and let $$\sigma({\mathcal P})$$ be the $$\sigma$$-ring generated by $$\mathcal P$$. Let further $$X$$ and $$Y$$ be (real or complex) Banach spaces with norms denoted by $$|\cdot|$$, let $$L(X,Y)$$ be the Banach space of all bounded linear operators from $$X$$ to $$Y$$ and let $$m:{\mathcal P}\to L(X, Y)$$ be an operator-valued measure which is countably additive in the strong operator topology. We define the semivariation $$\widehat m$$ of $$m$$ on $$\sigma({\mathcal P})\cup\{T\}$$ by the formula $\begin{split}\widehat m(A)= \sup\Biggl\{\Biggl|\sum^n_{i=1} m(A\cap A_i)x_i\Biggr|: A_i\in{\mathcal P},\;A_i\cap A_j=\emptyset\quad\text{if}\\ i\neq j,\;x_i\in X,\;|x_i|\leq 1,\;1\leq i\leq n,\;n=1,2,3,\dots\Biggr\}.\end{split}$ From now, we will assume that $$m$$ has finite semivariation on each $$A\in{\mathcal P}$$. For the term “$$m$$-a.e. on $$T$$”, we use as $$m$$-null sets the sets $$N\in\sigma({\mathcal P})$$ such that $$\widehat m(N)= 0$$. The Dobrakov integral in this setting is obtained in the following manner:
A simple $$m$$-integrable function $$s:T\to X$$ is any function of the form $s= \sum^n_{i=1} x_i\chi_{A_i},$ where $$x_i\in X\backslash\{0\}$$, $$A_i\in{\mathcal P}$$, $$A_i\cap A_j=\emptyset$$ for $$i\neq j$$, $$1\leq i,j\leq n$$, $$n=1,2,3,\dots\;$$. Its $$m$$-integral over a set $$A\in \sigma({\mathcal P})\cup\{T\}$$ is defined by the formula $\int_A s dm= \sum^n_{i=1} m(A\cap A_i)x_i.$ We denote by $${\mathcal I}_s(m)$$ the set of all simple $$m$$-integrable functions.
A function $$f:T\to X$$ is called $${\mathcal P}$$-measurable if there exists a sequence $$(s_n)$$ in $${\mathcal I}_s(m)$$ such that $$\lim_n s_n(t)= f(t)$$ for all $$t\in T$$.
A $${\mathcal P}$$-measurable function $$f:T\to X$$ is said to be $$m$$-integrable (in the sense of Dobrakov) if there exists a sequence $$(s_n)$$ in $${\mathcal I}_s(m)$$ converging $$m$$-a.e. on $$T$$ to $$f$$ and such that the sequence $$\left(\int_{(\cdot)} s_n dm\right)$$ is uniformly countably additive on $$\sigma({\mathcal P})$$. In this case, the $$m$$-integral of $$f$$ over a set $$E\in\sigma({\mathcal P})\cup \{T\}$$ is defined by the formula $\int_E fdm= \lim_n \int_E s_n dm.$ With these definitions, the survey paper in review is organized as follows. In section 5 are presented some basic properties of the Dobrakov integral. In section 6 are described the four pseudonormed spaces associated with $$m$$ and, in particular, the space $${\mathcal L}_1(m)$$. In section 7 are established the generalizations of the Lebesgue classical convergence theorems to $${\mathcal L}_1(m)$$. In section 8 is compared the Dobrakov integral with the abstract Lebesgue integral, the Bochner integral, the Pettis integral, the Bartle-Dunford-Schwartz integral, the Bartle bilinear and $$(*)$$ integrals and the Dinculeanu integral, putting emphasis in the advantage of the Dobrakov integral. In section 9 is defined the product of two operator-valued measures and is established a Fubini theorem with an important special case. Finally, in section 19 is presented the Radon-Nikodým theorem for the Dobrakov integral obtained by Maynard.

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