completion of L^2

In the book 'Lectures on Groups and Spaces For Physicists', Isham says
that the space L^2 is incomplete but that it can be completed by filling
in the gaps. I find this misleading. Isn't it the case that completing
L^2 is simply a matter of changing from the Riemann integral to the
Lebesgue integral - so in fact we don't need to 'add' anything in the
case? [But the Lebesgue integral is able to 'see' some (all??) of the
functions that the Riemann integral misses].
Thanks,
David.


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