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Your files have been transferred and they are on your MacBook Air, but you have to log in another user. Migration Assistant just told you that some of the apps you had are for PowerPC, and these apps aren't compatible with OS X Mountain Lion.
Go to > Log Out, and log in the new user that Migration Assistant created, so you will access to your migrated files. See > https://discussions.apple.com/docs/DOC-5472
I followed your instructions and found the normal user account as well as a Shared folder.
But when I opened the Shared folder, I only found files that cannot really be used:
I opened each of these files and they had folders such as "Application Support."
Maybe I'm just horrible at this stuff and the solution is staring me in the face!
You are right. The user was not transferred, or Migration Assistant created a mess, as I see folders "from old Mac".
Just run Migration Assistant again and untick "Applications" when Migration Assistant asks you for what you want to restore. It's clear that you have PowerPC applications, and they won't run
This is what it keeps saying when I use Migration Assistant:
Some of the software on your old computer is incompatible with this version of OS X. Because this software may cause problems with your computer, it was not transferred to your new computer:
Parallels: this version of Parallels is not compatible with this version of OS X.
For more information about incompatible software, see this Apple Support article: http://support.apple.com/kb/HT3258.
Is there an easy solution to this?