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The Completeness Problem in Spaces of Pettis Integrable Functions
Kazimierz Musial
Abstract
Two subspaces of the space of Banach space valued Pettis
integrable functions are considered: the space P(μ, X, var) of
Pettis integrable functions with integrals of finite variation in a Banach
space X and LLN (μ, X*, var) is always complete
and P(μ, X*, var) is complete if Martin's axiom and
the perfectness of
μ are assumed. Moreover,
a non-trivial example of a non-conjugate Banach space X with non-complete P(μ,
X, var) is presented.
Mathematics Subject Classification (2000): Primary: 46G10; Secondary:
28B05, 28A15.
Keywords: completeness, Pettis integral, lifting, vector measures,
measure preserving transformations, differentiation theory, differentiation
of set functions, vector valued integration, vector valued measures, Zermelo-Fraenkel,
axiom of choice, functional analysis, Hahn-Banach, Banach, banach space, Pettis,
Pettis integrable, complete
Quaestiones Mathematicaes 24 (4) 2001, 441–452
integrable functions are considered: the space P(μ, X, var) of
Pettis integrable functions with integrals of finite variation in a Banach
space X and LLN (μ, X*, var) is always complete
and P(μ, X*, var) is complete if Martin's axiom and
the perfectness of
μ are assumed. Moreover,
a non-trivial example of a non-conjugate Banach space X with non-complete P(μ,
X, var) is presented.
Mathematics Subject Classification (2000): Primary: 46G10; Secondary:
28B05, 28A15.
Keywords: completeness, Pettis integral, lifting, vector measures,
measure preserving transformations, differentiation theory, differentiation
of set functions, vector valued integration, vector valued measures, Zermelo-Fraenkel,
axiom of choice, functional analysis, Hahn-Banach, Banach, banach space, Pettis,
Pettis integrable, complete
Quaestiones Mathematicaes 24 (4) 2001, 441–452
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Quaestiones Mathematicae. ISSN: 1607-3606