On Vitali-Hahn-Saks-Nikodym type theorems

Barbara T. Faires

Annales de l'institut Fourier (1976)

  • Volume: 26, Issue: 4, page 99-114
  • ISSN: 0373-0956

Abstract

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A Boolean algebra 𝒜 has the interpolation property (property (I)) if given sequences ( a n ) , ( b m ) in 𝒜 with a n ≤ b m for all n , m , there exists an element b in 𝒜 such that a n ≤ b ≤ b n for all n . Let 𝒜 denote an algebra with the property (I). It is shown that if ( μ n : 𝒜 → X ) ( X a Banach space) is a sequence of strongly additive measures such that lim n μ n ( a ) exists for each a ∈ 𝒜 , then μ ( a ) = lim n μ n ( a ) defines a strongly additive map from 𝒜 to X and the μ n ' s are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive X -valued measures defined on 𝒜 is derived from the Nikodym boundedness theorem. A proof of the VHS theorem for group-valued measures is given.

How to cite

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Faires, Barbara T.. "On Vitali-Hahn-Saks-Nikodym type theorems." Annales de l'institut Fourier 26.4 (1976): 99-114. <http://eudml.org/doc/74304>.

@article{Faires1976,
abstract = {A Boolean algebra $\{\cal A\}$ has the interpolation property (property (I)) if given sequences $(a_n)$, $(b_m)$ in $\{\cal A\}$ with $a_n\le b_m$ for all $n,m$, there exists an element $b$ in $\{\cal A\}$ such that $a_n\le b\le b_n$ for all $n$. Let $\{\cal A\}$ denote an algebra with the property (I). It is shown that if $(\mu _n:\{\cal A\}\rightarrow X)$ ($X$ a Banach space) is a sequence of strongly additive measures such that $\lim _n\mu _n(a)$ exists for each $a\in \{\cal A\}$, then $\mu (a)=\lim _n\mu _n(a)$ defines a strongly additive map from $\{\cal A\}$ to $X$and the $\mu ^\{\prime \}_ns$ are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive $X$-valued measures defined on $\{\cal A\}$ is derived from the Nikodym boundedness theorem. A proof of the VHS theorem for group-valued measures is given.},
author = {Faires, Barbara T.},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {4},
pages = {99-114},
publisher = {Association des Annales de l'Institut Fourier},
title = {On Vitali-Hahn-Saks-Nikodym type theorems},
url = {http://eudml.org/doc/74304},
volume = {26},
year = {1976},
}

TY - JOUR
AU - Faires, Barbara T.
TI - On Vitali-Hahn-Saks-Nikodym type theorems
JO - Annales de l'institut Fourier
PY - 1976
PB - Association des Annales de l'Institut Fourier
VL - 26
IS - 4
SP - 99
EP - 114
AB - A Boolean algebra ${\cal A}$ has the interpolation property (property (I)) if given sequences $(a_n)$, $(b_m)$ in ${\cal A}$ with $a_n\le b_m$ for all $n,m$, there exists an element $b$ in ${\cal A}$ such that $a_n\le b\le b_n$ for all $n$. Let ${\cal A}$ denote an algebra with the property (I). It is shown that if $(\mu _n:{\cal A}\rightarrow X)$ ($X$ a Banach space) is a sequence of strongly additive measures such that $\lim _n\mu _n(a)$ exists for each $a\in {\cal A}$, then $\mu (a)=\lim _n\mu _n(a)$ defines a strongly additive map from ${\cal A}$ to $X$and the $\mu ^{\prime }_ns$ are uniformly strongly additive. The Vitali-Hahn-Saks (VHS) theorem for strongly additive $X$-valued measures defined on ${\cal A}$ is derived from the Nikodym boundedness theorem. A proof of the VHS theorem for group-valued measures is given.
LA - eng
UR - http://eudml.org/doc/74304
ER -

References

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