Decomposition of group-valued additive set functions
Annales de l'Institut Fourier, Tome 22 (1972) no. 3, p. 131-140
Soit $m$ une fonction additive définie sur un clan $H$ à valeurs dans un groupe topologique commutatif séparé et soit $K$ un idéal de $H$. On donne des conditions suffisantes pour que $m$ soit la somme de deux fonctions additives, l’une essentiellement portée sur $K$, l’autre nulle sur $K$. Ce résultat est utilisé pour obtenir deux décompositions de Lebesgue. On indique aussi d’autres applications ainsi que la théorie correspondante pour les mesures extérieures.
Let $m$ be an additive function on a ring $H$ of sets, with values in a commutative Hausdorff topological group, and let $K$ be an ideal of $H$. Conditions are given under which $m$ can be represented as the sum of two additive functions, one essentially supported on $K$, the other vanishing on $K$. The result is used to obtain two Lebesgue-type decomposition theorems. Other applications and the corresponding theory for outer measures are also indicated.
@article{AIF_1972__22_3_131_0,
author = {Traynor, Tim},
title = {Decomposition of group-valued additive set functions},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
address = {Gap},
volume = {22},
number = {3},
year = {1972},
pages = {131-140},
doi = {10.5802/aif.427},
zbl = {0228.28004},
mrnumber = {48 \#11439},
language = {en},
url = {http://www.numdam.org/item/AIF_1972__22_3_131_0}
}

Traynor, Tim. Decomposition of group-valued additive set functions. Annales de l'Institut Fourier, Tome 22 (1972) no. 3, pp. 131-140. doi : 10.5802/aif.427. http://www.numdam.org/item/AIF_1972__22_3_131_0/

[1] C.E. Rickart, Decomposition of additive set functions, Duke Math. Jour., 10 (1943), 653-665. | MR 5,232c | Zbl 0063.06492

[2] M. Sion, Outer measures with values in a topological group, Proc. Lond. Math. Soc. (3), 19 (1969), 89-106. | MR 39 #398 | Zbl 0167.14503

[3] M. Sion, Group-valued outer measures, International Congress of Mathematicians, Nice, 1970. | Zbl 0224.28008

[4] T. Traynor, Absolute continuity for group-valued measures (to appear), Can. Math. Bull., 1973. | MR 50 #7475 | Zbl 0289.28010

[5] T. Traynor, A general Hewitt-Yosida Decomposition, (to appear). Can. Jour. Math. | Zbl 0219.46034