2002 Ford Focus Dimming Deck lights/no start

I recently recieved your answer concerning my starter and I appreciate the advice. I have noticed for some time now that while driving and even sitting idle, my deck lights and headlights get very dim. In fact they almost seem to not be on. Then with a quick pump of the gas pedal (idle) or just a little more speed they come back to life. I can't help but feel that the voltage regulator is either on it's last leg or has gone already. This morning my low battery indicator came on but went off as fast as I looked at it. I am still having to occasionally push start the vehicle and was wondering if indeed it could be the alternator instead of the recently installed starter. I know that the voltage regulator is a common problem with these vehicles and is inside the alternator.I appreciate any advice you can offer.