The Oxford English Dictionary defines a peg as 'a drink, especially of spirits' and indicates that the term is mainly used within the UK & Commonwealth, rather than in the US.
So, as far as the OED is concerned, no specific measure is involved.
Wikipedia suggests that a 'peg' is (in most places) an informal measurement of between 1 and 2 fluid ounces:
http://en.wikipedia.org/wiki/Peg_(drinking)
However, it also points out that, in some parts of the world, the relevant authorities have fixed a standard measurement for a 'peg'.
Although neither of those sources mentions it, I'd suggest that offering someone a 'peg' of whiskey is synonymous with offering them a 'shot' of the spirit.
Chris