Generalized Fractal Strings Viewed as Measures

Abstract

In this chapter, we develop the notion of generalized fractal string, viewed as a measure on the half-line. This is more general than the notion of fractal string considered in Chapter 1 and in the earlier work on this subject (see the notes to Chapter 1). We will use this notion in Chapter 5 to formulate the explicit formulas which will be applied throughout the remaining chapters. Besides ordinary fractal strings, generalized fractal strings enable us to deal with strings whose lengths vary continuously or whose multiplicities are nonintegral or even infinitesimal. In Section 4.2, we discuss the spectrum of a generalized fractal string, and in Section 4.3, we briefly discuss the notion of generalized fractal spray, which will be used in Chapters 9 and 11.