  Let $f:[0,1]^d\to\mathbb{R}$ be a completely monotone integrand as defined by
Aistleitner and Dick (2015) 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 (NPLD)
property provides a computable lower bound. It has been known since Gabai
(1967) that the two dimensional Hammersley points in any base $b\ge2$ have
non-negative local discrepancy. Using the probabilistic notion of associated
random variables, we generalize Gabai's finding to digital nets in any base
$b\ge2$ and any dimension $d\ge1$ when the generator matrices are permutation
matrices. We show that permutation matrices cannot attain the best values of
the digital net quality parameter when $d\ge3$. 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 $d\ge2$ either fails to be projection regular or
has all its points on the main diagonal.
