A quantity $\int |E\mathscr{B} f| dP$ (or equivalently $\int|u - P(A: \mathscr{B})| dP, 0 < u < 1)$ associated with a $\sigma$-algebra $\mathscr{B}$ is shown to act as a criterion for a type of convergence of $\sigma$-algebras. This quantity also defines an ordering of $\sigma$-algebras, so that upper and lower limits can be defined in terms of this quantity. Another criterion for the convergence of $\sigma$-algebras is described based on the existence of these limits.
Publié le : 1974-02-14
Classification:
60-00,
Strong convergence of $\sigma$-algebras,
$L^1$-norm of conditional expectation,
Existence of upper and lower limits of $\sigma$-algebras,
60G20,
60A05,
28A05
@article{1176996752,
author = {Kudo, Hirokichi},
title = {A Note on the Strong Convergence of $\Sigma$-Algebras},
journal = {Ann. Probab.},
volume = {2},
number = {6},
year = {1974},
pages = { 76-83},
language = {en},
url = {http://dml.mathdoc.fr/item/1176996752}
}
Kudo, Hirokichi. A Note on the Strong Convergence of $\Sigma$-Algebras. Ann. Probab., Tome 2 (1974) no. 6, pp. 76-83. http://gdmltest.u-ga.fr/item/1176996752/