In some research papers, protease production has been represented by protease activity. So are both protease activity and protease production the same?
Not necessarily. Most proteases are produced as proenzymes that are catalytically inactive until a proteolytic cleavage cuts the inhibitory peptide. In that case, the enzyme preparation may be totally or partially inactive.
Additionally, any protein that is overexpressed may be produced partially or totally in an insoluble or aggregated form that has little or no activity.
Making the connection between protease concentration/production and its activity is often difficult or even impossible. Not only because the are produced in a pro-form but many other reasons like presence of inhibitors (for example TIMPs for MMPs).
Maybe you should give a look to our recent paper on the topic
There is a difference between both, sometimes there is a production of an enzyme but it is inactive either because it is produced in an inactive form (proform) or because of the presence of some inhibitors which prevent its activity. Therefore you can't guarantee if there is a high production of the enzyme this means there is an increase in its activity.
If there is no inhibitor unintentionally present in the protease preparation, you can determine the concentration of active protease by performing an active site titration. This requires a potent inhibitor. By comparing the concentration of active protease to the concentration of total protease, you can calculate the percentage of protease that is active.