I have the same problem, and unfortunately the solution as given above doesn't seem to work for me. I get the installer, but then the installer crashes in exactly the same way Logic does.
I have tried to extract the pkg-files using the program 'unpkg', then I get an error which is shorter than my Mac's error log:
Extracting "ProAudioCoreContent"...
Error while extracting archive:(ProAudioCoreContent.pkg/Payload): archived-checksum SHA1's do not match (No such file or directory)
gzip: /var/folders/nt/22yb35k159v28n7_q6ymkxyr0000gn/T/tmpZ83_1L/ProAudioCoreContent. pkg/Payload: invalid compressed data--format violated
cpio: (Empty error message)
cpio: (Empty error message)
This is more or less the same message I get from my error log; there seems to be some error in the checksum. I called Apple, they asked me to reinstall Lion from scratch. Which I did, then I didn't install anything but tried to install Logic Pro straight away, but to no avail; I get exactly the same error.
I am really at a loss; if this doesn't happen to everyone, it must have something to do with the hardware (I have a 2.4 GHz Intel Core 2 Duo MacBook Pro, from the beginning of 2008). I have sent my question and error log to Apple, they are looking into it, but I don't know how long that will take... Meanwhile I would like to record some songs! Ah well, back to good ol' GarageBand then...