Problem with character encoding ( danish æ ø å ) when using page handler include files in hook_menu().
The problem was, as Andy pointed out, that the file wasn't encoded in UTF-8.

The .inc file was created from the context menu in Windows: 'New' -> 'Text Document', which defaults to the ANSI encoding.

The solution is simply to convert the file to UTF-8, that can be done for example in Sublime Text by choosing 'File' -> 'Reopen with Encoding'. Other text editors have similar commands.