This chapter introduces signed and complex measures, focusing on integration with respect to these measures. The Radon-Nikodym theorems are proved, along with the study of Radon-Nikodym derivatives and their applications. Important topics include the Hahn and Jordan decompositions, absolute continuity of measures, and pointwise differentiation of measures. Integration of complex-valued functions and related concepts are also explored.

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Signed Measure and Complex Measure

  • Satya N. Mukhopadhyay,
  • Subhasis Ray

摘要

This chapter introduces signed and complex measures, focusing on integration with respect to these measures. The Radon-Nikodym theorems are proved, along with the study of Radon-Nikodym derivatives and their applications. Important topics include the Hahn and Jordan decompositions, absolute continuity of measures, and pointwise differentiation of measures. Integration of complex-valued functions and related concepts are also explored.