A Note on the Strong Convergence of $\Sigma$-Algebras
Copyright policy )A Note on the Strong Convergence of $\Sigma$-Algebras
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.
Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets, Axioms; other general questions in probability, Strong convergence of $\sigma$-algebras, $L^1$-norm of conditional expectation, Limit theorems in probability theory, Existence of upper and lower limits of $\sigma$-algebras, 60-00, 60G20, 60A05, 28A05
Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets, Axioms; other general questions in probability, Strong convergence of $\sigma$-algebras, $L^1$-norm of conditional expectation, Limit theorems in probability theory, Existence of upper and lower limits of $\sigma$-algebras, 60-00, 60G20, 60A05, 28A05
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