How Does One Measure Enthalpy?

One looks it up in tables. Try "Thermodynamic Properties of Fluids" by Mayhew & Rogers, or just about any ASHRAE psychrometric chart.

I conclude then that one does not measure enthalpy directly. Instead one will need to measure a dry bulb temperature and then also a wet bulb temperature (to determine % Relative Humidity). If the the inlet air pressure to a fan or air compressor is at sea level then I can use a sea level Psychometric Chart, but for discharge enthalpy, where the discharge air pressure is somewhat variable (it would depend on the actual discharge air pressure) then I would need to find (if one exists) a psychrometric chart that has pressure as a variable, and a high enough pressure value for the outlet of an air compressor.