Strange voltage reading on ungrounded system
It looks like the neutral side of the inverter is still connected to the bulb from your diagram. 

First off AC can travel across conductors and through air because there is a small amount of capacitance between, this turns parts of the circuit that are floating into something like an antenna.

If I plug my meter into the hot side (120V) of an outlet and let the other end wave in the air, I get a measurement of roughly 14-28V because the other probe functions something like an antenna. Something similar could be happening in your case because volt meters can detect very small currents, but impossible to say exactly what without knowing all the details of your setup. 

This is one of the reasons to have a ground, so you have something to reference voltages to. Floating pieces of metal (like the disconnected portion of your circuit) make bad references to make voltage measurements. A good idea would be to string a long cable back to the chassis of the inverter so you have something to reference to, or ground your system.