The first measurements of diboson production cross sections in proton-proton interactions at a center-of-mass energy of 5.02 TeV are reported. They are based on data collected with the CMS detector at the LHC, corresponding to an integrated luminosity of 302 pb^{-1}. Events with two, three, or four charged light leptons (electrons or muons) in the final state are analyzed. The WW, WZ, and ZZ total cross sections are measured as σ_{WW}=37.0_{-5.2}^{+5.5}(stat)_{-2.6}^{+2.7}(syst) pb, σ_{WZ}=6.4_{-2.1}^{+2.5}(stat)_{-0.3}^{+0.5}(syst) pb, and σ_{ZZ}=5.3_{-2.1}^{+2.5}(stat)_{-0.4}^{+0.5}(syst) pb. All measurements are in good agreement with theoretical calculations at combined next-to-next-to-leading order quantum chromodynamics and next-to-leading order electroweak accuracy.
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