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Math Help - uniform convergence

  1. #1
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    uniform convergence

    Let f_n= n \chi_{[\frac{1}{n},\frac{2}{n}]} \, on \mathbb{R} for all n \in \mathbb{N} and f=0 on \mathbb{R}.
    Show that ther does not exist a set of measure zero ,on the complement of which (f_n) is uniformly convergent.
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  2. #2
    Senior Member Tinyboss's Avatar
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    Quote Originally Posted by problem View Post
    Let f_n= n \chi_{[\frac{1}{n},\frac{2}{n}]} \, on \mathbb{R} for all n \in \mathbb{N} and f=0 on \mathbb{R}.
    Show that ther does not exist a set of measure zero ,on the complement of which (f_n) is uniformly convergent.
    Your functions f_n are converging pointwise to the zero function. If you had some set E as above, then you'd need, for any \varepsilon>0, that there exists some N_\varepsilon so that |f_n|<\varepsilon on E for all n>N_\varepsilon. But if E has measure zero, then the complement contains points arbitrarily near zero, i.e. points x such that x\in[\frac1n,\frac2n] for n as large as you like, which contradicts uniform convergence.
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