On the distinguishing features of the Dobrakov integral.

*(English)*Zbl 0883.28011The 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.

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.

Reviewer: P.Morales (Sherbrooke)

##### MSC:

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

46G10 | Vector-valued measures and integration |