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Coulomb interactions are crucial in determining the ground state of an ideal two-dimensional electron gas (2DEG) in the limit of low electron densities. In this regime, Coulomb interactions dominate over single-particle effects. In gallium arsenide, electrons are typically localized at these low densities. In contrast, in transition metal dichalcogenides, Coulomb correlations in a 2DEG can be anticipated at experimentally relevant electron densities. We focus on monolayer MoS₂. Equipping MoS₂ with gates allows electrons to be injected, creating a 2DEG. The electronic ground-state is probed optically with near-resonant photoluminescence spectroscopy as a function of the electron density. Owing to the robust optical selection, this probe is highly spin- and valley-selective. Crucially, the cross-polarized emission channels give information on intervalley exchange mechanisms. By identifying the formation of different trion states, we present experimental evidence of itinerant ferromagnetism in monolayer MoS₂. Out of the four available conduction bands, only two are occupied. These two bands have the same spin but different valley quantum numbers. This spin-polarization can be understood by strong intervalley Coulomb scattering in models including corrections to Fermi liquid theory.