ABSTRACT
The multinomial measure is a generalised version of the Cantor measure discussed in Example 1.2.1. For this example we derive expressions for both e(q) as in Equation 2.2, and the multifractal spectrum ef(y) as in Equation 2.7. We then show that e(q) and ef(y) are indeed related via the Legendre transform. This is done relatively simply using Lagrange multipliers. The Legendre transform also appears in the theory of large deviations, and in Part II of the book we show that this theory can be used to derive sufficient conditions for the Legendre transform relationships to hold in a more general setting.
