Most fuse checkers that I have looked at check the power by looking for the blown fuse, and will light up when there is a voltage drop across the fuse, in other words it will light up when the fuse is bad. There is no voltage drop across a good fuse, only a bad fuse.
My guess is the 40 amp fuse is tripped? If this is a automatic reset circuit breaker, then it will cool off and reset. If you have a inverter and factory installed converter/charger, you will need to unplug the charger before everything will work properly.
Also the inverter/charger should not be hooked up to the 40 amp circuit breaker, as this will trip the breaker when the inverter charger is drawing or charging at more than 40 amps. Most inverters will draw more than 40 amps, and will charge at 70 amps or so.
It would help to give the inverter specs if you have them. Was it recently installed = or factory installed?
Good luck!
Fred.
