  In this manuscript, we show how conformal invariance can be incorporated in a
classical theory of gravitation, in the context of metric measure space. Metric
measure space involves a geometrical scalar $f$, dubbed as density function,
which here appears as a conformal degree of freedom. In this framework, we
present conformally invariant field equations, the relevant identities and
geodesic equations. In metric measure space, the volume element and accordingly
the operators with integral based definitions are modified. For instance, the
divergence operator in this space differs from the Riemannian one. As a result,
a gravitational theory formulated in this space has a generalized second
Bianchi identity and a generalized conservation of energy-momentum tensor. It
is shown how, by using the generalized identity for conservation of
energy-momentum tensor, one can obtain a conformally invariant geodesic
equation. By comparison of the geodesic equations in metric measure space with
the Bohmian trajectories, in both relativistic and non-relativistic regimes, a
relation between density function $f$ and the quantum potential is proposed.
This suggests metric measure space to be considered as a suitable framework for
geometric description of Bohm's quantum mechanics. On the other hand, as it is
known, Weyl geometry is one of the main approaches to construct conformally
invariant gravitational models. Regarding the fact that the connection in the
integrable Weyl space is modified and in metric measure space remains the same
as it is in the Riemann space, the mathematical analogy between these two
spaces is also discussed.
