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Math Help - Lebesgue measurable

  1. #1
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    Lebesgue measurable

    Prove that if f:R->([0,∞]) and f^(-1)((r,∞]) ∈ M for each r∈ M,then f is
    Lebesgue measurable.
    (M is the σ-algebra of Lebesgue measurable sets)
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  2. #2
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    Re: Lebesgue measurable

    Quote Originally Posted by Sonifa View Post
    Prove that if f:R->([0,∞]) and f^(-1)((r,∞]) ∈ M for each r∈ M,then f is
    Lebesgue measurable.
    (M is the σ-algebra of Lebesgue measurable sets)
    I think you mean:

    Prove that if f:\mathbb{R} \to [0,\infty] and f^{-1}((r,\infty])\in \mathcal{M} for each r \in \mathbb{R}_{\ge 0}, then f is Lebesgue measurable.

    I am rewriting to verify what you are trying to prove because if r\in \mathcal{M}, then (r,\infty] does not make sense.

    Anyway, use subadditivity. (r_1,r_2] = (r_1,\infty]\setminus (r_2,\infty]. So, given any subset of [0,\infty], you can cover it with a countable union of sets of the form (r_1,r_2]. Then, think about the definition of the Lebesgue measure.
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