Converting MP3s to AIFF files

Do MP3 files contain all the musical information that is found in the AIFF files that are on CDs?

Since MP3s are a compression technology, I wonder if the information is still retained when converted back to AIFF.

Naturally this would be important to know if I'm downloading MP3s from Apple's Music Store and then burning a CD for playback though a high end system. Does anyone know the definitive answer to this?

When a 16bit/44.1MHz (redbook) signal is compressed to one of the MP3 formats information is thrown away. It cannot be recovered by converting the file format back to AIFF or WAV.
First of all, you're not downloading MP3s from the iTunes Store. You're downloading AACs, which are also compressed, but sound better than MP3s of the same bit resolution.

When you convert a compressed file (MP3 or AAC) back to WAV or AIFF, you are not going to get anything that sounds any better than the original compressed file. (It won't sound any worse, however.) The information that's lost during the compression stays lost. Still, you should try the AACs out, because they're really not bad for what they are.

I'd use iTunes to try out a few songs by an unfamiliar artist, or for material I'm only going to listen to on portable devices or in the car. But if I really liked something I downloaded, I'd go out and buy the CD to play on my "real" system.