I had a debate with my brother about this when I was young. It's a wonderful debate. I remember a point that if the Enterprise transporters managed to copy the entire of my brain, including all the memories, then from the point of view of the materialised 'new me' my historical memory would tell me I'm the same person.
I would remember walking up to the transporter pad, saying 'energise', and materialising at the destination. My memory is clear, and I am utterly convinced that that person before was the same me.
Then again, I look back now at some of my dim early memories, and realise how fallible memories are. I remember distinctly everyone saying 'Kindergarden' and read it as such, and only recently did I discover it was in fact 'Kindergarten'. How could I get it wrong, in particular a word so frequently used? I struggle to remember memories in early childhood - my grandmothers house, dim cloudy days and mist, a tornado? (I don't think it was, but at the time I thought I saw one). I know it is logical to think that my body was there experiencing things, but to be honest, 'I' wasn't there - I simply can't be where I can't remember being.
Add to this that if I took a prescription drug that alters the way I think, could you say that that was me? I suppose my conclusion is our experience is subjective, and paradoxically also not all of who we are. Evidence tells me that I'm more than my memories, and thus I agree with you @gwolffe356 - I wouldn't step on that transporter pad either.
About life extension though - that is also a scary thought. Imagine all the grumpy old men we know, and imagine them living for another 10,000 years. They would be very, very, very grumpy by then.