I'm misunderstanding something about the completeness of the space L^infinity(X); the space of functions from X into R such that the function is bounded everywhere except a null set.

This space is supposed to be complete, i.e. every Cauchy sequence converges inside L^infinity. But by the definition of the norm on L^infinity, applied to a Cauchy sequence seems to just say that eventually all functions in the sequence are bounded by the same number. But that says nothing about what happens to the function in-between that bound. So then many functions could be the limit.

Question

I'm misunderstanding something about the completeness of the space L^infinity(X); the space of functions from X into R such that the function is bounded everywhere except a null set.

This space is supposed to be complete, i.e. every Cauchy sequence converges inside L^infinity. But by the definition of the norm on L^infinity, applied to a Cauchy sequence seems to just say that eventually all functions in the sequence are bounded by the same number. But that says nothing about what happens to the function in-between that bound. So then many functions could be the limit.

anonymous · Answer

So then the limit doesn't need to exist.

I know the result has been proven by many different people. So I must be misunderstanding it.

anonymous · Answer

I've figured it out. I wasn't thinking properly about how the L^infinity norm of f_n - f_m can be arbitrary close to zero which means that all points must become arbitrarily close to each other.