Strongly inhomogeneous multifractal measures involving extremely diverse lo
cal probability scales and exhibiting left-sided multifractal spectra are c
onsidered. The standard multifractal formalism is extended to such drastica
lly inhomogeneous multifractal measures by introducing additional filtering
variables and scaling exponents, conjugated to these variables. It is show
n that the generalized multifractal formalism provides a more adequate and
complete description of complex structures of strongly inhomogeneous multif
ractal measures, in comparison with conventional formalism. (C) 2001 Elsevi
er Science B.V. All rights reserved.