A Nonabsolutely Convergent Integral Defined by Partitions of the Unity

In [5], a type of integral defined by partitions of the unity (PU-integral) is defined on an abstract compact metric measure space and it is proved that a PUintegrable function is μ−integrable and that the μ−integral is equivalent to the PU-integral. Moreover an example of a non Euclidean space on which is defined this type of integral is given. The PU-integral is obtained by approssimations of type Riemann sums. The advantage to use a such integral is that it does not use the geometry of the space so it can be defined in any abstract space. In this paper X denotes a compact metric space, M a σ-algebra of subsets of X such that each open set is in M, μ a non-atomic, finite, complete Radon measure on M such that: α) each ball U(x, r) centered at x with radius r has a positive measure, β) for every x in X there is a number h(x) ∈ < such that μ(U [x, 2r]) ≤ h(x) · μ(U [x, r]) for all r > 0 (where U [x, r]) is the closed ball), γ) μ(∂U(x, r)) = 0 where ∂U(x, r) is the boundary of U(x, r).