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Integral extension with locally-integral seminorm. (English) Zbl 0787.28006
For Loomis’ finitely additive ‘one-sided completion’ integral [L. H. Loomis, Am. J. Math. 76, 168-182 (1954; Zbl 0055.101)], a new definition is given and convergence theorems are obtained. If \(B\) is a vector lattice of real-valued functions on \(X\) and \(I: B\to\mathbb{R}\) linear non- negative (without any continuity assumptions), then the \(f: X\to\overline\mathbb{R}\) in question are characterized by the existence of \(h_ n\in B\) with \(I(| h_ n- h_ m|)\to 0\) and \(h_ n\to f\), \(I^*\)-locally, i.e., \(I^*(| f- h_ n|\land h)\to 0\) for each \(0\leq h\in B\); the set \(R_ 1(B,I)\) of these \(f\) is also the closure of \(B\) in \(\overline \mathbb{R}^ X\) with respect to Schäfke’s local integral norm \(I^*_ B\) [F. W. Schäfke, J. Reine Angew. Math. 289, 118- 134 (1977; Zbl 0337.28011)], where \(I^*(g):=\inf\{I(h): g\leq h\in B\}\), \(I^*_ B(g):=\sup\{I^*(g\land h): h\in B\}\).
Generalizing results of P. Bobillo Guerrero and M. Díaz Carrillo [Arch. Math. 52, No. 3, 258-264 (1989; Zbl 0674.28005)] one has \(R_ 1(B,I)\subset B_ 0+ \{f\in R_ 1(B,I): I^*_ B(| f|)=0\}\), \(B_ 0:=I\)-summable functions of \(B.+ C.\) (P. Bobillo Guerrero and M. Díaz Carrillo (loc. cit.)). In the special case \(\Omega\)=(semi)ring from \(X\), \(\mu: \Omega\to [0,\infty)\) finitely additive, \(B=\) real-valued stepfunctions \(h\) with respect to \(\Omega\), \(I(h):= \int hd\mu\), the \(R_ 1(B,I)\) coincides with the ‘abstract Riemann-\(\mu\)-integrable’ functions \(R_ 1(\mu,\overline\mathbb{R})\) of the reviewer [“Integration” (1985; Zbl 0568.28009), esp. p. 70 and 199], \(I^*\)-local convergence is ‘\(\mu\)-local convergence’, the convergence theorems for \(R_ 1(\mu,\overline\mathbb{R}\)) are special cases of those obtained here for \(R_ 1(B,I)\); if \(X\in\Omega\), \(R_ 1(B,I)\cap\mathbb{R}^ X= L(X,\Omega,\mu,\mathbb{R})\) of ’Dunford-Schwartz [N. Dunford and J. T. Schwartz, Linear operators. I (1958; Zbl 0084.104), esp. p. 112].
Reviewer: H.Günzler (Kiel)
MSC:
28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
28A25 Integration with respect to measures and other set functions
28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
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