Although initially LiveJournal’s "domain mapping" looked promising, all it actually appears to do is let me point my domain at it. While the first page presents itself as http://blog.filklore.com/, if you follow any link off that page, you find yourself on http://filklore-on-lj.livejournal.com/ again.
With Blogger, when you publish on a custom domain, it does the job completely. Not only does it accept page requests for a custom domain, but all links in the page served are for that custom domain. As a result, all pages are served within http://blog.filklore.com/
If you want to see what I mean, compare the following two pages
http://blog.filklore.com/ – pointing at my blog on Blogger
http://blog2.filklore.com/ – pointing at my blog on LiveJournal
and see what happens when you follow links, or even comment.
It may seem a fine point, and some folks may wonder why I am bothering. But if LiveJournal says it can handle custom domains (and I purchased a paid account to do it), it should do the job properly, in my opinion.