Signed Measure

This chapter is on signed measures. The Jordan–Hahn theorem decomposes a signed measure into the difference of two measures in a unique way. This result is crucial to prove the Radon–Nikodym theorem which generalizes the concepts of the derivative and the indefinite integral, to measures. It leads to the Lebesgue decomposition theorem which decomposes any measure on the Borel σ-field into sum of three measures. The three component measures being respectively, (i) absolutely continuous with respect to the Lebesgue measure, (ii) purely discrete, and (iii) singular with respect to the Lebesgue measure but without any discrete component.