We combine FUV spectra obtained with FUSE and STIS/E140M of HE 0226–4116 ($z=0.495$), PG 1116+215 ($z_{\rm em}=0.177$), and PG 1259+593 ($z_{\rm em}=0.478$) to determine the distribution of column density and Doppler width of the Ly$\alpha$ absorber population at $z<0.5$ detected in the spectra of these 3 quasars. The high spectral resolution UV spectra allow us to derive simultaneously the redshift, column density, and Doppler width of each absorber. Two different populations of H I absorbers appear to be present: narrow and broad H I absorptions that could be tracers of the warm photoionised IGM ($T < 10^5$ K) and warm-hot ionised medium (WHIM, $T \sim 10^5-10^6$ K). For a reliable assessment of the baryonic content of the Ly$\alpha$ forest at low $z$ it is crucial to determine the relative numbers of photoionised absorbers vs. hot collisionally ionised absorbers. Preliminary results indicate that 30–40% of the total baryons are in the photoionised IGM and at least 20% in the WHIM. We also discuss the observed evolution of the Doppler parameter with redshift.