It is still a challenge to assess the merger fraction of galaxies at different cosmic epochs in order to probe the evolution of their mass assembly. Using the Illustris cosmological simulations, we investigate the relation between the separation of galaxies in a pair, both in velocity and projected spatial separation space, and the probability that these interacting galaxies will merge in the future. From this analysis, we propose a new set of criteria to select close pairs of galaxies along with a new corrective term to be applied to the computation of the galaxy merger fraction. We then probe the evolution of the major and minor merger fraction using the latest MUSE deep observations over the HUDF, HDFS, COSMOS-Gr30 and Abell 2744 regions. From a parent sample of 2483 galaxies with spectroscopic redshifts, we identify 366 close pairs spread over a large range of redshifts ($0.2<z<6$) and stellar masses ($10^7-10^{11}M_{odot}$). Using the stellar mass ratio between the secondary and primary galaxy as a proxy to split the sample into major, minor and very minor mergers, we found a total of 183 major, 142 minor and 47 very minor close pairs corresponding to a mass ratio range of 1:1-1:6, 1:6-1:100 and lower than 1:100, respectively. Due to completeness issues, we do not consider the very minor pairs in the analysis. Overall, the major merger fraction increases up to $zapprox 2-3$ reaching 25% for pairs with the most massive galaxy with a stellar mass $M^*geq 10^{9.5}M_{odot}$. Beyond this redshift, the fraction decreases down to $sim 5$% at $zapprox 6$. The evolution of the minor merger fraction is roughly constant with cosmic time, with a fraction of 20% at $z<3$ and a slow decrease between $3leq z leq6$ to 8-13%.
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