Alexander Pupyshev's suggestion for measurement of degradation of labeled BSA through endocytic trafficking is good. A complementary method to this is the long-lived protein degradation assay which gives quantitative information about the capacity for degradation of long-lived proteins in the lysosome trafficked through autophagy. The assay requires radioactive 14C-valine and the possibility of measuring radioactivity of samples. Let me know if you want a detailed protocol. 

An alternative to look at degradation through the endocytic pathway is to follow the degradation of the EGF-receptor after addition of EGF. 

There are also more indirect assays to quantify the amount and activity of lysosomal proteins, like the "Magic Red" cathepsin B flourescence or measuring NAG activity in lysates, however these approaches don't actually give direct information about degradation.

It is an old questionn without simple answer. Because lysosomes are not a simple machine. The main estimation of its functional activity is connected with intralysosomal digestion of  their enzyme substrates. We adressed labeled bovine serum albumin into lysosomes by endocytosis and followed for release of degradation product from intact isolated lysosomes incubated in standard conditions. This estimation is originated from researches of John Mego from USA decades ago.  Another approach is indirect and is connected with determination of inner pH because it reflects an ability of lysosomes to do their  digestive work as whole and in general to fuse with inflowing vesicles. Neutral inner pH of lysosomes blocks lysosomal fusion ability.  Methodically there is a lot of detectors of intralysosomal pH (acridine orange, neutral red, various Lysotrackers etc.) Finally, many agents can induce permeabilization of  lysosomal membrane that is an apoptogenic change. The alteration can be detected by the both aproaches  noted above plus by  determination of non-sedimental activity of lysosomal enzymes, its elevation...

Alexander Pupyshev's suggestion for measurement of degradation of labeled BSA through endocytic trafficking is good. A complementary method to this is the long-lived protein degradation assay which gives quantitative information about the capacity for degradation of long-lived proteins in the lysosome trafficked through autophagy. The assay requires radioactive 14C-valine and the possibility of measuring radioactivity of samples. Let me know if you want a detailed protocol. 

An alternative to look at degradation through the endocytic pathway is to follow the degradation of the EGF-receptor after addition of EGF. 

There are also more indirect assays to quantify the amount and activity of lysosomal proteins, like the "Magic Red" cathepsin B flourescence or measuring NAG activity in lysates, however these approaches don't actually give direct information about degradation.

Good suggestions above re dyes... Also, follow link below to a great paper for some ideas to look at cathepsin B/L activities and lysosomal acidity: http://www.nature.com/cr/journal/v23/n4/full/cr201311a.html

However, for simplicity, I would recommend looking at the autophagosome marker LC3-II, with and without Bafilomycin A1 (20 nM, 2-4 hours). This assay should give a robust read-out of autophagic-lysosomal flux. If your compound inhibits lysosomal proteolysis, you will see a large accumulation of steady-state LC3-II by western blot (16kDa) and no flux in comparison to control.