There's more than one way. The way I do is using
Google Apps for my domain. That entails moving the email handling to Gmail.
The other way I can think of, and that a lot of people use, entails keeping mail at LP and forwarding it to a Gmail account.
Or you can use Gmail to fetch LP mail via POP.
With the latter two options, if you want to send from your domain email address, Outlook users will see this:
From:
username@gmail.com on behalf of Your Name [
user@domain.com]
Which would be your preferred method? Are there any other requirements? This would affect the answer to your question.