Before doing anything, switch the power off and test before placing your hand on any of the wires. Some very poorly connected wires can still carry electricity even when switched off! So, be extremely careful.
First question: why?
It's simple, if you ever used a similar ceiling light like the one below, you will know why. This is some very old technology, you have a transformer inside, you have a starter, and if you have circular big tube light inside. If any one of them gone wrong, it won't even turn on. Frequent failure is the prime motive! Nowadays when energy saving light bulb is so cheap and popular, it does not make sense to use this kind of old technology any more!
As you can see in the picture below, I have highlight a screw in which the transformer is attached. You actually have two screws like this, one on each side, and you can use any one of those for fixing the light bulb socket.
First of all, remove all those old technologies:-
Then attached the light bulb socket there in any one of the fixing screws, since they're originally decided to fix a heavy transformer, the small socket with the small light bulb is definitely fine:-
Connect the wires, nothing can go wrong since it's alternating current:-
Done! It's now so simple and elegant, you have a single light bulb to go wrong. You can switch to a higher power one to have a brighter room, you can easily replace damaged bulb, and it's instant on as well!