We report on precise Doppler measurements of L231-32 (TOI-270), a nearby M dwarf (d = 22 pc, M⋆ = 0.39 M⊙, R⋆ = 0.38 R⊙), which hosts three transiting planets that were recently discovered using data from the Transiting Exoplanet Survey Satellite (TESS). The three planets are 1.2, 2.4, and 2.1 times the size of Earth and have orbital periods of 3.4, 5.7, and 11.4 d. We obtained 29 high-resolution optical spectra with the newly commissioned Echelle Spectrograph for Rocky Exoplanet and Stable Spectroscopic Observations (ESPRESSO) and 58 spectra using the High Accuracy Radial velocity Planet Searcher (HARPS). From these observations, we find the masses of the planets to be 1.58 ± 0.26, 6.15 ± 0.37, and 4.78 ± 0.43 M⊕, respectively. The combination of radius and mass measurements suggests that the innermost planet has a rocky composition similar to that of Earth, while the outer two planets have lower densities. Thus, the inner planet and the outer planets are on opposite sides of the 'radius valley' - a region in the radius-period diagram with relatively few members - which has been interpreted as a consequence of atmospheric photoevaporation. We place these findings into the context of other small close-in planets orbiting M dwarf stars, and use support vector machines to determine the location and slope of the M dwarf (Teff < 4000 K) radius valley as a function of orbital period. We compare the location of the M dwarf radius valley to the radius valley observed for FGK stars, and find that its location is a good match to photoevaporation and core-powered mass-loss models. Finally, we show that planets below the M dwarf radius valley have compositions consistent with stripped rocky cores, whereas most planets above have a lower density consistent with the presence of a H-He atmosphere.