I was under the impression that Google already did this. A Googler already described this to me years ago as "closing the loop", where Google's ad network already exposed a customer to a brand/product, but Google only gets credit for the conversion if it happens in the same browser session online. By tying together stats between impression and conversion (purchase, even in-store), the ad network becomes more valuable…
PPC Ad-words perhaps influences me if I'm researching a product or service, but that's mostly a function of search results placement rather than . Usually I scan down below the first few search results because those are people trying to tell me they're what I'm looking for and they usually aren't.
I will readily admit that I have to consider that I am a freakish outlier, though.