  In string theory, the consequences of replacing the measure of integration
$\sqrt{-\gamma}d^2 x$ in the Polyakov's action by $\Phi d^2 x$ where $\Phi$ is
a density built out of degrees of freedom independent of the metric
$\gamma_{ab}$ 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.
