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Math Help - f is L(A) implies finite almost everywhere?

  1. #1
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    f is L(A) implies finite almost everywhere?

    LET f is Lebesgue Integrable on A where m(A) is finite, Prove that f must be finite almost everywhere on A.

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    Re: f is L(A) implies finite almost everywhere?

    \left\{f=\pm\infty\right\}=\bigcap_{n\geq 1}\left\{|f|\geq n\right\} and m(\left\{|f|\geq n\right\})\leq \frac 1n\int_A|f|dm. Since m(A)<\infty, m(\left\{f=\pm\infty\right\})=\lim_{n\to\infty}m( \left\{|f|\geq n\right\}) and you can conclude.
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    Re: f is L(A) implies finite almost everywhere?

    Quote Originally Posted by girdav View Post
    \left\{f=\pm\infty\right\}=\bigcap_{n\geq 1}\left\{|f|\geq n\right\} and m(\left\{|f|\geq n\right\})\leq \frac 1n\int_A|f|dm. Since m(A)<\infty, m(\left\{f=\pm\infty\right\})=\lim_{n\to\infty}m( \left\{|f|\geq n\right\}) and you can conclude.
     \bigcap_{n\geq 1}\left\{|f|\geq n\right\} ?? or union??
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    Super Member girdav's Avatar
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    Re: f is L(A) implies finite almost everywhere?

    Quote Originally Posted by younhock View Post
     \bigcap_{n\geq 1}\left\{|f|\geq n\right\} ?? or union??
    No, it's the intersection, since if |f(x)|=+\infty then |f(x)| in greater than each integer.
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    Re: f is L(A) implies finite almost everywhere?

    Quote Originally Posted by girdav View Post
    No, it's the intersection, since if |f(x)|=+\infty then |f(x)| in greater than each integer.
    Oh yes i get this. But why is m(\left\{|f|\geq n\right\})\leq \frac 1n\int_A|f|dm ?
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    Super Member girdav's Avatar
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    Re: f is L(A) implies finite almost everywhere?

    n\cdot m(\left\{|f|\geq n\right\})\leq \int_{\left\{|f|\geq n\right\}}|f|dm\leq \int_{A}|f|dm
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    Re: f is L(A) implies finite almost everywhere?

    Quote Originally Posted by girdav View Post
    n\cdot m(\left\{|f|\geq n\right\})\leq \int_{\left\{|f|\geq n\right\}}|f|dm\leq \int_{A}|f|dm
    OKAY!! THANKS a lot!!!!!
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    Super Member girdav's Avatar
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    Re: f is L(A) implies finite almost everywhere?

    And note that the result is true for a \sigma-finite measured space (X,\mathcal A,\mu), i.e. a space such that we can find a countable partition of X into sets of finite measure (for example \mathbb R with Lebesgue measure).
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