i do not know if you try it,but put battery inside,put usb cable in ,connect it to PC ..(1st try to use usb 3.0) and maybe your camera will switch on..idea is based on that some model of Fuji camera have inside firmware option to choose what type of battery is used..as if selection is set to wrong type,then camera will not switch on. Well,to pass that we will connect camera to pc ,to see if respond ,and even after some time usually can be can be turned on in many cases,or even batteries can be charged(depends of choice of batteries), but really is worth to try to see if camera will respond when is connected to pc..
So its not lamp..check one by one,from power outlet ,then cords,switch...some floor halogen lamp have power safety switch(some halogen floor lamp have it),if in outlet have power,what will be 1st thing..then if power came to as you wrote ,bulb..simple screwdriver power tester is enough for that.. So if power is on outlet then came to cord and not came to bulb,you need to dismantle,and look if from cord came to switch,and then if when switch is closed("ON")if pass the switch to came to bulb or other shape or type of lighting element. Sometimes bulb is not perfect manufactured and not make good contact,even is screwed tight.. If you have dimmer on lamp,check that too.(dimmer can be bypassed,and lamp can work without..but when dimmer is burned ,can prevent power to came to bulb.)