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Abstract Let $$f:[0,1]^{d}\to{\mathbb{R}}$$ be a completely monotone integrand as defined by Aistleitner and Dick (2015, Acta Arithmetica, 167, 143–171) and let points $$\boldsymbol{x}_{0},\dots ,\boldsymbol{x}_{n-1}\in [0,1]^{d}$$ have a non-negative local discrepancy (NNLD) everywhere in $$[0,1]^{d}$$. We show how to use these properties to get a non-asymptotic and computable upper bound for the integral of $$f$$ over $$[0,1]^{d}$$. An analogous non-positive local discrepancy property provides a computable lower bound. It has been known since Gabai (1967, Illinois J. Math., 11, 1–12) that the two-dimensional Hammersley points in any base $$b\geqslant 2$$ have NNLD. Using the probabilistic notion of associated random variables, we generalize Gabai’s finding to digital nets in any base $$b\geqslant 2$$ and any dimension $$d\geqslant 1$$ when the generator matrices are permutation matrices. We show that permutation matrices cannot attain the best values of the digital net quality parameter when $$d\geqslant 3$$. As a consequence the computable absolutely sure bounds we provide come with less accurate estimates than the usual digital net estimates do in high dimensions. We are also able to construct high-dimensional rank one lattice rules that are NNLD. We show that those lattices do not have good discrepancy properties: any lattice rule with the NNLD property in dimension $$d\geqslant 2$$ either fails to be projection regular or has all its points on the main diagonal. Complete monotonicity is a very strict requirement that for some integrands can be mitigated via a control variate.