I bought this sconce light, which I am attaching to an outlet
controlled by a dimmer switch.

The electrical outlet gets the 110V as expected before attaching the
light.
After attaching the light, I put my multimeter probes into the socket
where the bulb should go and can see the 110 V. But when I put the
bulb in, it does not light up

I thought that the socket inside might be pushed all the way back and
hence bulb not making contact, but I don't think thats the case.
Interestingly, when I put in a fluorescent bulb instead, it just has a
small flicker going.

Any ideas ?