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We suggest commutation relations for a quantum measure. In one version ofthese relations, the right-hand side takes account of the presence of curvatureof 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 constantcurvature and show that in this case the commutation relations for the quantummeasure are analogues of commutation relations in loop quantum gravity. It isassumed that, in contrast to loop quantum gravity, a triangulation of space isa necessary trick for quantizing such a nonlocal quantity like a measure; indoing so, the space remains a smooth manifold. We consider the self-consistentproblem of the interaction of the quantum measure and classical gravitation. Itis shown that this inevitably leads to the appearance of modified gravities.Also, we consider the problem of defining the Euler-Lagrange equations for amatter field in the background of a space endowed with quantum measure.

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