Molecular clouds form from the atomic phase of the interstellar medium. However, characterizing the transition between the atomic and the molecular interstellar medium (ISM) is a complex observational task. Here we address cloud formation processes by combining H I self absorption (HISA) with molecular line data. Column density probability density functions (N-PDFs) are a common tool for examining molecular clouds. One scenario proposed by numerical simulations is that the N-PDF evolves from a log-normal shape at early times to a power-law-like shape at later times. To date, investigations of N-PDFs have been mostly limited to the molecular component of the cloud. In this paper, we study the cold atomic component of the giant molecular filament GMF38.1-32.4a (GMF38a, distance = 3.4 kpc, length ~ 230 pc), calculate its N-PDFs, and study its kinematics. We identify an extended HISA feature, which is partly correlated with the ¹³CO emission. The peak velocities of the HISA and ¹³CO observations agree well on the eastern side of the filament, whereas a velocity offset of approximately 4 km s⁻¹ is found on the western side. The sonic Mach number we derive from the linewidth measurements shows that a large fraction of the HISA, which is ascribed to the cold neutral medium (CNM), is at subsonic and transonic velocities. The column density of the CNM part is on the order of 10²⁰ to 10²¹ cm⁻². The column density of molecular hydrogen, traced by ¹³CO, is an order of magnitude higher. The N-PDFs from HISA (CNM), H I emission (the warm and cold neutral medium), and ¹³CO (molecular component) are well described by log-normal functions, which is in agreement with turbulent motions being the main driver of cloud dynamics. The N-PDF of the molecular component also shows a power law in the high column-density region, indicating self-gravity. We suggest that we are witnessing two different evolutionary stages within the filament. The eastern subregion seems to be forming a molecular cloud out of the atomic gas, whereas the western subregion already shows high column density peaks, active star formation, and evidence of related feedback processes.