Description:
Definitions for measures on Boolean algebras.
Content:
Definitions for measures on Boolean algebras.
Definition: Assume that \(\cal B = (B,\land,\lor,{\bf{1}},{\bf{0}})\) is a Boolean algebra.
- A measure or additive real valued measure on \(\cal B\) is a function \(m:\cal B\to {\Bbb R}\) such that \(m({\bf{1}})=1\),\((\forall x\in B)(m(x) \ge 0)\), and \((\forall x,y\in B)(x\land y ={\bf{0}} \to m(x\lor y) = m(x) + m(y))\). (It follows that\(m({\bf{0})} = 0\).)
- A two valued measure on \(\cal B\) is a measure on \(\cal B\) whose range is \(\{0,1\}\).
- If \(\cal B = (\cal P(X),\cap,\cup,X,\emptyset)\) then a measure \(m\) on \(\cal B\) is non-principal if \(m(\{x\}) = 0\) for every \(x\in X\).
Howard-Rubin number:
147
Type:
Definitions
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