I know about the VA/Watts thing, but it doesn't belong at the OP's level of understanding.
I think it may go to the lamp too. However, there MAY be another explanation of why no voltage. Interlocks. the lamp going open or shorted for that matter makes more sense. I have seen lamps short. The headlight on one of the cars I owned did, resulting in blinking headlights every 20 seconds or so.
It might be wise to check the resistance of the lamp. HP ratings for a high power lamp makes sense too and not the motor. I have seen and operated one of these things in grade school, but I may have been 10-13 Years old.
FWIW to the OP: Suppose you have a switch(relay), fuse, power and a lamp. The voltage across the switch should be the line voltage when the lamp is off an 0V when the lamp is ON. When you know the circuit configuration, that can be a better test. The voltage ACROSS a switch when the load is on is a good indication of the health of the switch.
ASIDE: If you had the lamp out and measured the voltage across the (not lamp) and got 120 and with the switch off, you got 0. Then when you put the lamp in and got 0 Volts when it's supposed to be on and 0 when its supposed to be off, you could have a bad switch.