Here are three ideas for answering this question:
- Take a camcorder and turn it on so it is recording. Then put it in the refrigerator, close the door and open it again. You can rewind and watch the tape to see if the light turned off.
- If you want to be lower tech, take a camera with a self-timer and stick it in so it shoots the picture (no flash) while the door is closed.
- If you want to be higher tech, you could install a Webcam in your refrigerator and watch it live on the Internet.
Another idea would be to put a thermometer on the bulb and leave the door shut for 10 minutes. Then open the door and check the thermometer immediately. If it is cold, then you know the bulb was off when the door was closed.
Here are some interesting links: