Quantum measure as a necessary ingredient in quantum gravity and modified gravities

We suggest commutation relations for a quantum measure. In one version of these relations, the right-hand side takes account of the presence of curvature of space; in the simplest case, this yields the action of general relativity. We consider the cases of the quantization of the measure on spaces of constant curvature and show that in this case the commutation relations for the quantum measure are analogues of commutation relations in loop quantum gravity. It is assumed that, in contrast to loop quantum gravity, a triangulation of space is a necessary trick for quantizing such a nonlocal quantity like a measure; in doing so, the space remains a smooth manifold. We consider the self-consistent problem of the interaction of the quantum measure and classical gravitation. It is shown that this inevitably leads to the appearance of modified gravities. Also, we consider the problem of defining the Euler-Lagrange equations for a matter field in the background of a space endowed with quantum measure.