In the conventional two-measure theory, the scalar density function $\Phi$ is taken to be $\Phi \equiv {ϵ}^{\mu \nu \rho \sigma }{ϵ}_{abcd}({\partial }_{\mu }{\phi }^a)({\partial }_{\nu }{\phi }^b)({\partial }_{\rho }{\phi }^c)({\partial }_{\sigma }{\phi }^d)$, where the indices $a,b,c,d=1$, 2, 3, 4 are internal-space indices. It is more natural to replace the four scalars ${\phi }^a$ by a Lorentz-covariant four-vector ${\phi }^m$ with a local Lorentz index $m=(0)$, (1), (2), (3). We entertain this possibility, and show that the newly proposed Lagrangian respects not only Lorentz covariance, but also global-scale invariance. The crucial equation ${\partial }_{\mu }L=0$ in the conventional two-measure theory also arises in our new formulation, as the ${\phi }^m$-field equation.

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