I have looked over the proofs that show convergence a.e. imply convergence in measure. I understand the proofs, but I do not understand why one must go into such detail.
It seems as though one builds a series out of the values of x where the series (of functions) do not converge, and then demonstrates that the limit of the series has measure zero.
But, convergence almost everywhere has a subset built-in with measure zero. Why is it not possible to simply show that where the function does not converge, the measure is zero (since this is the definition of a.e. convergence)?
Thank you. (Sorry there is nothing explicate written here, I will look up the formatting to write things properly tomorrow).