how could measure whether or not the chemical reaction that breaks down hydrogen peroxide is occurring

You could check the pH!

Explanation:

This is an idea, when you break down H2O2 (Hydrogen peroxide) you will be left with H2O (Water) and O2 (Oxygen gas). H2O2 theoretically would have a lower pH than water. Water has an average pH of around 7 while H2O2 will have a pH of about 4.5.

So theoretically to figure out if hydrogen peroxide is being broken down you can test for the pH. This is a very simply process, you just stick a little peice of litmus paper into the solution and immediatly it will change color to tell you the pH.

sorry to break it to ya but no one can answer your question it's just

jk but i am serious about the part that most people cant answer it beacuse you did not mention the section you are talking about. but it would if you added some detail and were a bit more specific.