<abstract xmlns="http://www.w3.org/1999/xhtml"><p> Let <italic>(X,A,</italic>μ<italic>)</italic> be a complete probability space, ρ a lifting, <italic>T</italic>ρ the associated Hausdorff lifting topology on X and <italic>E</italic> a Banach space. Suppose <italic>F:</italic> (X,Tρ<italic>)</italic><italic>-> E”</italic>σ be a bounded continuous mapping. It is proved that there is an <italic>A</italic> ∈ <italic>A</italic> such that <italic>FXA</italic> has range in a closed separable subspace of <italic>E</italic> (so <italic>FXA:X</italic> → <italic>E</italic> is strongly measurable) and for any <italic>B</italic> ∈ <italic>A</italic> with μ<italic>(B)</italic> > 0 and B ∩ <italic>A =</italic> ø, <italic>F<sub>XB</sub></italic> cannot be weakly equivalent to a <italic>E-</italic>valued strongly measurable function. Some known results are obtained as corollaries.</p></abstract>

Let (X,A,μ) be a complete probability space, ρ a lifting, Tρ the associated Hausdorff lifting topology on X and E a Banach space. Suppose F: (X,Tρ)-> E”σ be a bounded continuous mapping. It is proved that there is an A ∈ A such that FXA has range in a closed separable subspace of E (so FXA:X → E is strongly measurable) and for any B ∈ A with μ(B) > 0 and B ∩ A = ø, FXB cannot be weakly equivalent to a E-valued strongly measurable function. Some known results are obtained as corollaries.

A decomposition of bounded, weakly measurable functions

Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, U S.A.

Tatra Mountains Mathematical Publications

{"article-title":"A decomposition of bounded, weakly measurable functions"}

Let (X,A,μ) be a complete probability space, ρ a lifting, Tρ the associated Hausdorff lifting topology on X and E a Banach space. Suppose F: (X,Tρ)-> E”σ be a...

A decomposition of bounded, weakly measurable functions

Published Online: Nov 13, 2012

Page range: 67 - 70

DOI: https://doi.org/10.2478/v10127-011-0025-1

Keywords
liftings, weakly measurable functions, weakly equivalent functions, vector measures with finite variations.

This content is open access.

A decomposition of bounded, weakly measurable functions

Published Online: Nov 13, 2012

Page range: 67 - 70

DOI: https://doi.org/10.2478/v10127-011-0025-1

Keywordsliftings, weakly measurable functions, weakly equivalent functions, vector measures with finite variations.

This content is open access.

Keywords
liftings, weakly measurable functions, weakly equivalent functions, vector measures with finite variations.