I know a lot of you are probably like why the eff do you get lifesteal. Well simple, even though Lux is not ad based and her basic attacks dont do a lot of damage early game, getting just a little life steal can keep you in lane a lot longer and get you fed via minions. Its an edge up that many people dont think about.
Lichbane gives you a good dose of AP plus magic resistance. A lot of times Lux will go mid against another AP character and the magic resistance while minimal can make a big difference early game.
Ionion boots are better for Lux that Sorercors shoes in my opinion because the magic pen that you get from Sorcerors isnt as significant as the cooldown you can get from Ionion boots. This can bring your ult down to a 30 second cooldown after fully leveled.
Rush tear of the goddess because at this point in the game mana will be your best friend. This gives you uber mana regen with your masteries and runes. (well once you have a full rune page ><) PLUS Archangels staff is AMAZING when it comes to mana intense AP characters like Lux. Its passive not only gives you extra mana but bases your AP off of the mana, how sweet is that? pretty freaking sweet I say.
Last but certainly not least is Rabadons deathcap. Aptly names in my opinion because it certainly brings death. This will boost your AP enormously. It is the icing on the cake for a hardcore AP build.
SIDENOTE:
If you are winning and have your full build I always either build off of the vampiric sceptor, get some spell vamp items and if you are super awesome amazing you can always ditch the boots and stack an archangels or a rabadons.
That is all. Obviously feel free to get items in your own order if you think mine sucks but I feel like these items together are pretty sweet :)