There's already been some discussion here about how the boundaries of London and the rest of the south-east are somewhat fluid in some viewers'/listeners' minds - and the anomalies of terrestrial coverage - so hopefully I'm not repeating myself on this particular point...
BBC South East did a documentary in 2011 about the influence of Vox amplifiers on the British music scene - Vox Pop: How Dartford Powered The British Beat Boom (
http://www.bbc.co.uk/programmes/b017zwq8 - you can find it on YouTube too).
But until it got picked up by BBC4, it had only been aired on BBC1 South East - while Dartford itself is, at least for TV, strictly BBC London territory.
I've also seen events such as the Biggin Hill air fair also get more coverage on BBC South East than London, even though Biggin Hill is actually inside Greater London.
I'm not sure how far London Live's coverage goes for satellite and cable viewers - I know some viewers beyond the GL boundary get it on their cable providers, even though it'll never cover where they live.