Many solar phenomena display a scaling symmetry associated with random multiplicative cascades.
Here a physical measure, initially uniform on a spatial, temporal, or space-time set, is divided among subsets according to randomly determined fractions. This division is repeated on smaller and smaller sub-subsets, so that the resulting measure at the smallest scale is given at any point by the product of a string of random fractions comprising its fragmentation history. Such measures are highly intermittent. They characterize such solar phenomena as the spatial distribution of magnetic flux in an active region and the time distribution of global X-ray emission. For example, all PDFs which allow for zero fractions lead to measures with local peaks of unlimited strengths which are progressively less and less space filling.