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Math Help - Continuous function => Measurable

  1. #1
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    Continuous function => Measurable

    If a function f:[a,b]->R is continuous a.e, then show that f is a measurable function.
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  2. #2
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    Quote Originally Posted by Chandru1 View Post
    If a function f:[a,b]->R is continuous a.e, then show that f is a measurable function.
    The definition of continuity is that  f^{-1} takes open sets to open sets. f is a measurable function if  f^{-1} takes open sets to measurable sets. But, every open set is measurable.

    Thus, all that remains is to show that the a.e. condition doesn't effect measurability.

    Define g:[a,b]-->R to be:

    g(a) = f(a) if f is continuous at a
    g(a) = \lim_{x->a}{f(x)} if f is not continuous at a

    Then g is a continuous function, so g is measurable. But g = f a.e. thus, f is also measurable.


    Note: You may have to show g is well-defined
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  3. #3
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    Quote Originally Posted by southprkfan1 View Post
    The definition of continuity is that  f^{-1} takes open sets to open sets. f is a measurable function if  f^{-1} takes open sets to measurable sets. But, every open set is measurable.
    Isn't a function measurable iff the preimage of measurable sets is measurable?
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