Problem: Let A be a σ-algebra of subsets of a set X, and let M be a Banach space of bounded operators on a Hilbert space H. Consider a map : AM that assigns each set A E A a bounded operator μ(A) Є M, and suppose μ satisfies the conditions for an operator measure. Prove that the map μ defines a bounded operator-valued measure, and show that the integral of a bounded measurable function f : XC with respect to μ can be written as: x f(x) du(x) = f(A) dE(A), . σ(μ) where E is the corresponding spectral measure associated with the operator-valued measure μ. Solution Outline: 1. Operator-Valued Measures: Define operator measures and explain the properties they satisfy, including additivity and boundedness. 2. Spectral Representation: Use the spectral theorem for operator-valued measures to show that such measures can be represented in terms of spectral integrals. 3. Functional Calculus: Connect the operator-valued measure to functional calculus and demonstrate how the integral can be computed using the spectral representation. 4. Measurable Functions and Integration: Show how the integral of a function with respect to an operator measure can be interpreted in terms of the spectral measure E.

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