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The main objective of this book is to give a collection of criteria available in the spectral theory of selfadjoint operators, and to identify the spectrum and its components in the Lebesgue decomposition. at the total spectral measure associated with it;often studying such ameasure meant looking at some transform of the measure. The transforms were of the form f,?(A)f which is expressible, by the spectral theorem, as ?(x)dµ (x) for some finite measure µ . The two most widely used functions were the sx ?1 exponential function?(x)=e and the inverse function?(x)=(x?z) . These functions are “usable” in the sense that they can be manipulated with respect to addition of operators, which is what one considers most often in the spectral theory of Schrodinger type operators. Starting with this basic structure we look at the transforms of measures from which we can recover the measures and their components in Chapter 1. In Chapter 2 we repeat the standard spectral theory of selfadjoint op- ators. The spectral theorem is given also in the Hahn–Hellinger form. Both Chapter 1 and Chapter 2 also serve to introduce a series of definitions and notations, as they prepare the background which is necessary for the criteria in Chapter 3.