We study the transformed measures with respect to the real parameters of a self-similar measure on a self-similar Cantor set to give a simple proof for some result of its multifractal spectrum. A transformed measure with respect to a real parameter of a self-similar measure on a self-similar Cantor set is also a selfsimilar measure on the self-similar Cantor set and it gives a better information for multifractals than the original self-similar measure. A transformed measure with respect to an optimal parameter determines Hausdorff and packing dimensions of a set of the points which has same local dimension for a self-similar measure. We compute the values of the transformed measures with respect to the real parameters for a set of the points which has same local dimension for a self-similar measure. Finally we investigate the magnitude of the local dimensions of a self-similar measure and give some correlation between the local dimensions.
