When writing a novel how would one best represent background dialogue from a radio or television that is perhaps only half heard by the characters but still important for the reader? Should I have the narrator state clearly that the conversation or broadcast is not being heard by the characters but sprinkle in key phrases and passages important for the reader and should they be written in italics?
I would say this is highly dependent on just how important you want what is being said to be to the reader. For the hypothetical, lets suggest that the important story is the famous quote and it was totally serious:
"My fellow Americans, I'm pleased to tell you today that I've signed legislation that will outlaw Russia forever. We begin bombing in five minutes." -Ronald Reagan
Scenario 1: The scene is built around suggesting the period or routine, but bits will become important now.
You would write it with the narrative voice discussing the Radio reporting on legislation the President signed about outlawing Russia and his plans to enact those laws. Later on the news story will be brought up and given contexts by the characters as it becomes more important to what the actual report is saying. Since it's not important to the scene at the moment, it doesn't need to interrupt the action of the scene but does need to be shown. Narrative writers are unique in that if it's unusual enough to mention, it's going to stand out in the readers mind... especially if you call back to it later in your story.
Scenario two: It changes the topic of the scene.
If the scene is not initially about the subject of the report, but the report changes the narrative topic of the current scene, then you would basically write the same as above and then have some cue to the fact that an important news story has just started going on... perhaps one character announces that he or she wants to hear the story (the classic scene in the dinner where a patron yells "Hey, turn it up!" as it starts or an indication that the sound of the "BREAKING NEWS" jingle is interrupting the current fluff piece...). This follows by the story... You should probably get a little artsy with the dialog as it needs to indicate that the speaker is not able to respond to the people in the scene, but how is up to you... just be consistent. If it's from the TV, give a description of either the studio and before the dialog or better yet, just go to the initial picture on the scene and have the anchor narrate over it or throw to a journalist on the scene... normally in breaking news events, the picture is limited to just a straight shot of the scene and a voice over (usually on phone, especially in modern times) talking about what is being seen... if radio, ignore this. Either way, I prefer to write my news story dialog in a box quote (format to center justified and make the rows of text line up near equally where possible... like a newspaper article, even if it's not news paper). If you need to describe action, do a hard cut to the dialog with a dash (-) or an ellipse (...) and describe the event happening in regular text format and return to the box quote to continue the story... This includes the studio anchor talking to the field reporter, but start them as regular dialog cuts.
As a bit of advice for realistic on the scene reporting of breaking news, I highly recommend watching some 9/11 as-it-happened-coverage (most major news networks have uploaded from initial report to collapse of the second tower on YouTube, in complete or in segment parts. NBC's Today Show coverage is especially good for a study as Today Show Anchor Matt Laurer interrupts mid-interview of an author of a new book to announce that there is a major news update, but then is told they are going to commercial to get everyone in place and ready to do Breaking News. After the one commercial is done, they are focused on the smoking tower.). No matter which coverage you watch, all of them were focused on the burning North Tower when the second plane strikes the South Tower and captures the news crew's live reactions to the very sudden change in events.
Scenario 3: Usual News story
If the report is on the President's well known by now statement, then you might want to do it a bit more like a narrative in a block text, but only for the obvious story piece (most news is anchor tossing to a reporter, who has a couple of words and then gives a prepared video report which is cut. This is basically a dialoged news story and in fact, most local news organizations, which use this format, will usually have on their web site a word for word article of the audio component of the film segment that reads like a newspaper article, so use the block quote style for the news story.
I would set it off into its own lines and use italics. Since it will happen multiple times, the reader will assume that it's background and characters may or may not hear what's said (or even notice it).
If you want to be clear who does and doesn't hear certain lines, you'll have to say so.
Luzia looked up with a start. "Russia? We're bombing Russia?"
"Mmmmmuh?" Geraldo put down his pen. "What?"