Plug the lamp into another wall plug to make sure there is not a problem with the wall socket itself. Screw a new light bulb into the lamp just in case the old bulb was burned out.
Remove the light bulb. Unscrew the socket and remove it. Twist the upper section of the lamp off to access the dimmer switch
Inspect the switch's leads first. If they are loose, reconnect them. If they are intact, test them with the multimeter. Carefully plug the lamp in. Set the multimeter to the voltage setting with a range to cover both 120-volt and 220-volt sockets. On most multimeters, this range is "1,000 volts." Connect the meter leads to the leads of the switch. If nothing registers on the meter at all it means the switch needs to be replaced.
Unscrew the screws holding the old switch in place and the screws holding the switch wires in place. Remove the old switch. Place the new switch in, tighten down its wires to the appropriate terminals on the lamp and then tighten the switch back into place.
Screw the top half of the lamp back on, install the socket and screw the light bulb in by hand. Plug in the lamp to test the new switch.