Mandelbrot, BenoitRiedi, Rudolf H.2007-10-312007-10-311997-01-202004-01-14B. Mandelbrot and R. H. Riedi, "Inverse Measures, the Inversion formula, and Discontinuous Multifractals," <i>Advances in Applied Mathematics,</i> 1997.https://hdl.handle.net/1911/20081Journal PaperThe present paper is part I of a series of three closely related papers in which the inverse measure m' of a given measure m on [0,1] is introduced. In the first case discussed in detail, both these measures are multifractal in the usual sense, that is, both are linearly self-similar and continuous but not differentiable and both are non-zero for every interval of [0,1]. Under these assumptions the Hölder multifractal spectra of the two measures are shown to be linked by the inversion formula f'(a) = a f(1/a) . The inversion formula is then subjected to several diverse variations, which reveal telling details of interest to the full understanding of multifractals. The inverse of the uniform measure on a Cantor dust leads us to argue that this inversion formula applies to the Hausdorff spectrum even if the measures m and m' are not continuous while it may fail for the spectrum obtained by the Legendre path. This phenomenon goes along with a loss of concavity in the spectrum. Moreover, with the examples discussed it becomes natural to include the degenerate Hölder exponents 0 and infinity in the Hölder spectra. This present paper is the first of three closely related papers on inverse measures, introducing the new notion in a language adopted for the physicist. Parts II and III make rigorous what is argued with intuitive arguments here. Part II extends the common scope of the notion of self-similar measures. With this broader class of invariant measures part III shows that the multifractal formalism may fail.engTemporaryMultifractalsJournal articleTemporaryhttp://dx.doi.org/10.1006/aama.1996.0500