  We prove a formula relating the Hausdorff dimension of a subset of the unit
interval and the Hausdorff dimension of the same set with respect to a random
path matric on the interval, which is generated using a multiplicative cascade.
When the random variables generating the cascade are exponentials of Gaussians,
the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum
gravity appears. This note was inspired by the recent work of Duplantier and
Sheffield proving a somewhat different version of the KPZ formula for Liouville
gravity. In contrast with the Liouville gravity setting, the one dimensional
multiplicative cascade framework facilitates the determination of the Hausdorff
dimension, rather than some expected box count dimension.
