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.