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Standard general coordinate invariance for the volume element is extended togeneral coordinate transformations that have a negative jacobian. This ispossible by introducing a non Riemannian Measure of integration, whichtransforms according to the jacobian of the coordinate transformation, not theabsolute value of the jacobian of the coordinate transformation as it is thecase with $\sqrt{-g}$. For some signed general transformations a change ofboundary conditions is involved in GR and restoring the original limits ofintegrations can restore symmetry of the action. The discussion totally changeswhen a non Riemannian Measure of integration is introduced, in the case ofsigned general coordinate transformations, and even when we have, boundaries,since the modified measure is constructed out of scalar fields becomes also theintegration manifold, Since the measure fields that define the non metricmeasure are scalars. if a signed general coordinate transformations isconsidered, there is no change in the measure fields, indicating strictinvariance of both lagrangian density and integration manifold, implyingboundary terms for manifolds with boundaries in coordinate space to beirrelevant . This analysis can be applied to give a framework where to certaininteresting scenarios . We consistently formulate the non Riemannian measuretheory extension of General Relativity, that could be related to Linde UniverseMultiplication model, although there are some fundamental differences. Theassumption of a coordinate independent measure of integration appears alsouseful in the study of Baby Universe Creation. The formulation provides also anew natural way to address the problem of time in Quantum Cosmology, a proposalfor a quantum space time and identification of space times and anti space timesstates in the gravitational theory.

Signed Coordinate Invariance, invariant lagrangians and manifolds, the  time problem in quantum cosmology, quantum space time, spacetimes and  antispacetimes