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    Difference Between Measurable Functions

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    Given: and and is measurable and is a null set.

    1) Is zero except on the null set, true of false?

    2) Where is a null set, true or false?

    I have been told that both 1 and 2 are true, but I don't understand why. Since the set {x: f(x)=g(x)} is null, it seems to me that d(x) should be only be zero on the null set, whereas apparently it is zero everywhere EXCEPT the null set. It may be a typo, but more likely I'm just not seeing something incredibly obvious. Also, I'm told that the truth of 1 implies 2, but again I don't see why.

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    Solution Preview

    Both 1 and 2 are false.

    1. False.
    Since the set {x: f(x)=g(x)} is a null set and the null set is the empty set. It means that for any x in E,
    f(x) is not ...

    Solution Summary

    Measurable functions are investigated in this solution.