Thank you for the replies. Is the fixture easy to replace, given that I have pretty much no experience in electrical / wiring work?
Yes, it's pretty simple. First turn of the breaker which powers that light. If you don't have it clearly identified in the panel, you're going to have to go by trial and error and test the power (using a multimeter or test light) until you find the right one. Even if your breaker panel is well labeled, you should test the confirm that the power is off.
Technically you could do this procedure by just turning off the switch. If you're comfortable working around electricity and the small chance of a shock, that's fine. But, if the original installer miswired the circuit (e.g. put the switch across neutral instead of hot), you could still have hot wires when you think it's off. The only way to know for sure is to turn off the breaker. That said, if you always treat wires like they may be hot, you'll be fine (and a 110V shock isn't that bad anyway IMHO).
Then remove the fixture. In the junction/ceiling box above the fixture should be the wire nuts connecting the fixture leads to the supply wires. After you unscrew them, set the old fixture aside. Now is a good time to make sure the supply wires are right. Make sure all three supply wires (black, white, bare/green) are separated and turn the breaker back on. With the light switch turned on, test the voltage:
- bare/green to black: 110-120V (or test light lit)
- white to black: 110-120V (or test light lit)
- bare/green to white: 0V (although this could be nonzero for various technical reasons, it should not be 110-120V).
Repeat the test with the light switch off:
- bare/green to black: 0V
- white to black: 0V
- bare/green to white: 0V
If you don't get the correct results, post back here with what you saw. If it all tests out OK, turn the breaker back off and install your new fixture.
Match the fixture wire colors to the supply wire colors and use the new wire nuts which were probably supplied with the fixture. You should not be able to see any bare conductor (except, obviously, if you have a bare ground wire in your supply) once you screw on the wire nut. It should just be finger tight, don't use pliers.
Remount the fixture, turn on the breaker, and test to ensure the light works.
Another thing I've noticed is that the light fixture in the living room (next to the kitchen), which uses incandescent light bulbs, also seems to burn out the bulbs much faster than it should. Is it possible that there is a bigger wiring problem in the apartment that's affecting both light fixtures?
Possibly. The most likely reason the incandescent bulbs fail quickly is the bulb is getting intermittent power due to a bad socket or loose connection. Turn off the switch, remove the bulb, confirm power is off, and bend the metal tab at the bottom of the socket out a bit. This will ensure you get a good contact with the bulb. When you reinstall the bulb, don't overtighten it. If you have the time, you could also check the wires as above and ensure the wire nuts are securely screwed on and making good contact between the three pairs of wires.
Matt