So a vaccinated person exposed to measles can contract the disease. Here’s the difference though - your immune system would recognize the pathogen, attack it and ramp up antibody production to fight the pathogen you were exposed to. You might not ever feel sick or experience any symptoms or you might get a very mild illness. This is the beauty of immunity. Yes, you could potentially contract a very mild case and be contagious though. 

As an immune compromised individual, I just want to say thanks for vaccinating yourself and your kid. You will reduce the risk of this disease spreading.