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Diffuse hydrogen gas has typically been observed in H\u2009i Lyman-\u03b1 (Ly\u03b1) absorption in the spectra of bright background quasars. However, it is important to extend these measurements to background galaxies as quasars become increasingly rare at high redshift and rarely probe closely separated sight lines. Here, we use deep integral field spectroscopy in the MUSE eXtremely Deep Field to demonstrate the measurement of the Ly\u03b1 transmission at z \u2248 4 in absorption to a background galaxy at z\u00a0= 4.77. The H\u2009i transmission is consistent with independent quasar sight lines at similar redshifts. Exploiting the high number of spectroscopic redshifts of faint galaxies (500 between z\u00a0= 4.0\u20134.7 within a radius of 8 arcmin) that are tracers of the density field, we show that Ly\u03b1 transmission is inversely correlated with galaxy density, i.e. transparent regions in the Ly\u03b1 forest mark underdense regions at z \u2248 4. 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