1. Introduction  
2. Data  
2.1. Summary of the ALPINE survey  
2.2. The [C II] emission line measurements  
3. Methodology  
3.1. Luminosity function calculation  
3.2. Accounting for [C II] non-detections  
4. Results and discussion  
4.1. Measurements based on the ALPINE UV sample  
4.3. Comparison with indirect observational expectations  
4.5. Implications to the molecular gas mass density at \( z \sim 4-6 \)  
5. Summary and conclusions