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Math Help - Existence of Real-Valued Functions Satisfying Certain Properties.

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    Existence of Real-Valued Functions Satisfying Certain Properties.

    (1)Is there a real-valued function f satisfies
    the set {(x,f(x)),x belongs to R} is a second category subset of R2?
    (2)Is there a real-valued function f satisfies
    the set {(x,f(x)),x belongs to R} is non-measurable in the Lebesgue sense?
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    Re: a question about real-valued functions on R

    Quote Originally Posted by mathabc View Post
    Is there a real-valued function f satisfies the set {(x,f(x)),x belongs to R} is non-measurable in the Lebesgue sense?
    Hint: For any A\subset \mathbb{R} non Lebesgue measurable, choose the indicator function \chi_A:\mathbb{R}\to \mathbb{R}

    \chi_A (x)=\begin{Bmatrix} 1 & \mbox{ if }& x\in A\\0 & \mbox{if}& x\not \in A\end{matrix}

    Is there a real-valued function f satisfies the set {(x,f(x)),x belongs to R} is a second category subset of R2?
    Hint: Use the Baire's Category Theorem: Every complete metric space is of second category.
    Last edited by FernandoRevilla; August 3rd 2011 at 08:59 AM.
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    MHF Contributor FernandoRevilla's Avatar
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    Re: a question about real-valued functions on R

    ... and \Gamma (I)=\Delta=\{(x,x):x\in\mathbb{R}\} is complete.

    And with this, we end the lively conversation.
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    Re: a question about real-valued functions on R

    Quote Originally Posted by FernandoRevilla View Post
    Hint: For any A\subset \mathbb{R} non Lebesgue measurable, choose the indicator function \chi_A:\mathbb{R}\to \mathbb{R}

    \chi_A (x)=\begin{Bmatrix} 1 & \mbox{ if }& x\in A\\0 & \mbox{if}& x\not \in A\end{matrix}
    Just an observation: The graph of that function is a subset of a set of measure 0 (the line y=1).
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    MHF Contributor FernandoRevilla's Avatar
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    Re: a question about real-valued functions on R

    Quote Originally Posted by Jose27 View Post
    Just an observation: The graph of that function is a subset of a set of measure 0 (the line y=1).
    Right, which implies \mu^* (\Gamma (1_A))=0 if you consider the complection \mu^* of \mu .

    P.S. I suppose you meant a subset of the union of the lines y=0 and y=1 .
    Last edited by FernandoRevilla; August 6th 2011 at 07:59 PM.
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