According to the principle of locality in physics, events taking place at different locations should behave independently of each other, a feature expected to be reflected in the measurements. In quantum field theory, measuring often requires to renormalize, so one expects renormalisation to preserve locality. However, with the commonly used regularisation methods, scuh as dimensional regularization, a "naïve finite part procedure" does not do the job so more sophisticated methods such as algebraic renormalisation à la Connes and Kreimer are implemented in order to preserve locality. We shall show how one can implement a "naïve finite part procedure" provided one uses a multivariate regularisation instead of a monovariate one such as dimensional regularisation. A multivariate regularisation yields a way to keep track of independence of events reflected in the fact that the corresponding meromorphic functions in several variables involve independent sets of variables. This multivariate renormalisation scheme, which takes place on the target space of meromorphic germs, can be applied to various situations involving renormalisation, such as Feynman integrals, multizeta functions and their generalisations, namely discrete sums on cones and discrete sums associated with trees.