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Thread: Null Sets and uniform convergence

  1. #1
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    Null Sets and uniform convergence

    Prove that if a sequence of continous functions uniformly converges on [a,b] then the union of their graphs is a null set.
    In other words:
    Prove that if the sequence $\displaystyle f_n:[a,b] \to R$ converges uniformly on [a,b] then the set : $\displaystyle A=(x,f_n) | x \in [a,b],n \in N$ is a null set...


    I know the function f is continous ( $\displaystyle f_n \to f $ ) and that the graph of the function f and of each $\displaystyle f_n $ if a null set....
    Can't figure out how to prove what I need to prove


    Thanks in advance
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  2. #2
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    Quote Originally Posted by WannaBe View Post
    Prove that if a sequence of continous functions uniformly converges on [a,b] then the union of their graphs is a null set.
    In other words:
    Prove that if the sequence $\displaystyle f_n:[a,b] \to R$ converges uniformly on [a,b] then the set : $\displaystyle A=(x,f_n) | x \in [a,b],n \in N$ is a null set...


    I know the function f is continous ( $\displaystyle f_n \to f $ ) and that the graph of the function f and of each $\displaystyle f_n $ if a null set....
    Can't figure out how to prove what I need to prove


    Thanks in advance
    Try a proof by contradiction.

    $\displaystyle P\rightarrow Q\equiv P\wedge$ ~$\displaystyle Q$
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  3. #3
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    ...

    Quote Originally Posted by dwsmith View Post
    Try a proof by contradiction.

    $\displaystyle P\rightarrow Q\equiv P\wedge$ ~$\displaystyle Q$
    If the set A isn't a null set, then there is an $\displaystyle \epsilon >0$ such as there is no finite amount of rectangles that cover the set A and their sum is less than $\displaystyle \epsilon$ .
    In other words, we assume by contradiction that there is an $\displaystyle \epsilon >0$ such as every finite amount of rectangles that cover the set A- their sum is $\displaystyle \geq \epsilon $ ...

    I've no idea how I can continue from this point... It seems to be very difficult to continue from this point...

    Hope you'll be able to help me

    Thanks !
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