We present a study of the galaxy population in the cluster RX J1053.7+5735 , one of the most distant X-ray selected clusters of galaxies , which also shows an unusual double-lobed X-ray morphology , indicative of a possible equal-mass cluster merger . The cluster was discovered during the ROSAT deep pointings in the direction of the Lockman Hole . Using Keck-DEIMOS spectroscopic observations of galaxies in the 2 \aas@@fstack { \prime } 0 \times 1 \aas@@fstack { \prime } 5 region surrounding RX J1053.7+5735 , we secured redshifts for six galaxies in the range 1.129 < z < 1.139 , with a mean redshift < z > =1.134 . The mean redshift agrees well with the cluster X-ray redshift previously estimated from the cluster X-ray Fe-K line , confirming the presence of a cluster at z \sim 1.135 . Galaxies with concordant redshifts are located in both eastern and western sub-clusters of the double cluster structure , indicating that both sub-clusters are at similar redshifts . This result is also consistent with a previous claim that both eastern and western X-ray lobes have similar X-ray redshifts . Based on their separation of \sim 250 kpc/h , these results support the interpretation that RX J1053.7+5735 is an equal-mass cluster merger taken place at z \sim 1 , although further direct evidence for dynamical state of the cluster is needed for a more definitive statement about the cluster merging state . The six galaxies have a line-of-sight velocity dispersion \Delta v \sim 650 km s ^ { -1 } . All six galaxies show clear absorption features of CaII H & K , and several Balmer lines , typical of early galaxies at the present epoch , in agreement with their I - K colors . A color-magnitude diagram , constructed from deep optical/NIR observations of the RX J1053.7+5735 field , shows a clear red color sequence . There is an indication that the red sequence in RX J1053.7+5735 lies \sim 0.3 to the blue of the Coma line , qualitatively consistent with previous studies investigating other clusters at z \sim 1 .