Abstract
Let \(\Omega =\{ r: r \in [0,1] \cap \mathbb{Q}\}\) denote the set of all rational numbers inside the interval [0,1], let \(\mathcal{A}\) be the algebra of sets that can be expressed as finite unions of non-intersecting sets A of the form {r: a < r < b}, {r: a ≤ r < b}, {r: a < r ≤ b}, or {r: a ≤ r ≤ b}, and let \(\mathsf{P}(A) = b - a\). Prove that the set-function P(A), \(A \in \mathcal{A}\), is finitely additive but not countably additive.
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Shiryaev, A.N. (2012). Mathematical Foundations of Probability Theory. In: Problems in Probability. Problem Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3688-1_2
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