In string theory, the consequences of replacing the measure of integration √-γ d2 x in the Polyakov action by Φ d2 x, where Φ is a density built out of degrees of freedom independent of the metric γab defined in the string, are studied. The string tension appears as an integration constant of the equations of motion. The string tension can change in different parts of the string due to the coupling of gauge fields and point particles living in the string. The generalization to higher-dimensional extended objects is also studied. In this case there is no need of a fine-tuned cosmological term, in sharp contrast to the standard formulation of the generalized Polyakov action for higher-dimensional branes.