For superstrings, the consequences of replacing the measure of integration √-γd2x in the Polyakov's action by ψd2x, whereis a density built out of degrees of freedom independent of the metric γab defined in the string, are studied. As in the Siegel reformulation of the Green-Schwarz formalism the Wess-Zumino term is the square of supersymmetric currents. As opposed to the Siegel case, the compensating fields needed for this do not enter into the action just as in a total derivative. They instead play a crucial role in making up a consistent dynamics. The string tension appears as an integration constant of the equations of motion. The generalization to higher dimensional extended objects is also studied using in this case the Bergshoeff and Sezgin formalism with the associated additional fields, which again are dynamically relevant unlike the standard formulation. Also unlike the standard formulation, there is no need for a cosmological term on the world brane.