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Measurability of a function f which is discontinuous only on a set of measure 0.

  1. Oct 3, 2011 #1
    1. The problem statement, all variables and given/known data

    Let f:[a,b] -> R be a bounded function, and let D be the set of points at which f is not continuous.

    (a) Prove that D is a countable union of closed sets.

    (b) Prove that if m(D) = 0, then f is measurable.


    2. Relevant equations

    Of(x) = lim(ε->0)(sup f(y) - inf f(y) ) on {y:|y-x|< ε} [Called the "Oscillation at x"]


    3. The attempt at a solution

    I know that if m(D) = 0, then f is Reimann integrable. This probably leads to a fact about Lebesgue integrability, which leads to a fact about measurability...? I'm not sure how it all ties together, or what the formal argument would be.

    Any help would be much appreciated : )
     
  2. jcsd
  3. Oct 3, 2011 #2
    (a) Let M be the upper bound for f & consider the sets
    O_n={x in D |2M/(n+1)<=Of(x)<=2M/n} n=1,2,..
    Do you see why each O_n must be closed & their union is D?

    (b)If D is a null set,f would be riemann integrable(& hence measurable) as it is bounded .
     
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