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Math Help - Measure theory

  1. #1
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    Measure theory

    Let f be an extended real valued function defined on a measurable set E of R. Show that f is a measurable function on E iff for each rational number r the set E(f<r)= \{ x \in E \ : \ f(x)<r\} is a measurable subset of E.
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  2. #2
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    Quote Originally Posted by Chandru1 View Post
    Let f be an extended real valued function defined on a measurable set E of R. Show that f is a measurable function on E iff for each rational number r the set E(f<r)= \{ x \in E \ : \ f(x)<r\} is a measurable subset of E.
    -->Suppose f is measurable

    then  f^{-1} takes opens sets to measurable sets

    Fix a rational number r

    But E(f<r)= \{ x \in E \ : \ f(x)<r\} = f^{-1}( -\infty, r), which is measurable

    <--Suppose for each rational number r the set E(f<r)= \{ x \in E \ : \  f(x)<r\} is a measurable subset of E.

    Fix b in  R

    We want to show f^{-1}( -\infty, b) is measurable.

    This is clearly true if b is rational (by assumption)

    If b is irrational, then we can find a sequence of rationals, {  q_n } that are increasing (or decreasing) and converge to b.

    Let  S_k = E(f <  q_k )
    Let S = E(f < b)

    Clearly,  S_1 \subset S_2 \subset ... and  S_k --> S, and since each  S_k is measurable then so is S
    Last edited by southprkfan1; April 7th 2010 at 11:08 AM.
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