Step 1: Separate the items by function: does each one act as a protective barrier, or does it merely measure exposure?
Step 2: Goggles guard the eyes, gloves guard the hands, and a lab coat guards the body and clothing, so all three are protective barriers and qualify as PPE.
Step 3: A radiation badge belongs to the category of personnel monitoring devices. It records dose for safety surveillance but offers no shielding, so it is the option that is not PPE. \[\boxed{\text{Badges for detecting radiation}}\]