Lesson plan
Ba space
In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma . The norm is defined as the variation, that is \|\nu\|=|\nu|(X). If Σ is a sigma-algebra, then the space ca(\Sigma) is defined as the subset of ba(\Sigma) consisting of countably additive measures. The notation ba is a mnemonic for bounded additive and ca is short for countably additive. If X is a topological space, and Σ is the sigma-algebra of Borel sets in X, then rca(X) is the subspace of ca(\Sigma) consisting of all regular Borel measures on X.
Article summary
In mathematics, the ba space ba(\Sigma) of an algebra of sets \Sigma is the Banach space consisting of all bounded and finitely additive signed measures on \Sigma . The norm is defined as the variation, that is \|\nu\|=|\nu|(X). If Σ is a sigma-algebra, then the space ca(\Sigma) is defined as the subset of ba(\Sigma) consisting of countably additive measures. The notation ba is a mnemonic for bounded additive and ca is short for countably additive. If X is a topological space, and Σ is the sigma-algebra of Borel sets in X, then rca(X) is the subspace of ca(\Sigma) consisting of all regular Borel measures on X.
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
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- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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