I have a similar problem, except my multimeter is showing 10~ volts at the socket.
When I plug in the light pole, I even get 10~ volts at the light bulb socket at the top.
However, when I plug a bulb in, I never get it to light up. I even bought a new pack of bulbs and none of them light up.
I even hooked the light bulb up to my 18v ryobi battery with some spare wire and the bulb lit up.
What gives!? I'm getting power to the socket, I've tried several bulbs... I cannot think of what else it could possibly be.