Let on for all and on

Show that ther does not exist a set of measure zero ,on the complement of which is uniformly convergent.

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- Mar 11th 2010, 07:45 AMproblemuniform convergence
Let on for all and on

Show that ther does not exist a set of measure zero ,on the complement of which is uniformly convergent. - Mar 11th 2010, 12:52 PMTinyboss
Your functions are converging pointwise to the zero function. If you had some set as above, then you'd need, for any , that there exists some so that on for all . But if has measure zero, then the complement contains points arbitrarily near zero, i.e. points such that for as large as you like, which contradicts uniform convergence.