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Real Analysis : Measurable Sets and Functions

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September 23rd, 2022

1).If f: X–> C ( C is complex plane) is measurable, then prove that f^-1({0}) ( f inverse of 0 or any other point) is a measurable set in X.

2). If E is measurable set in X and if

X_E ( x) = { 1 if x is in E, 0 if x is not in E} then X_E is a measurable function. Now I want you to prove the other direction, that is, I want you to show that
If X_E(x) is measurable, then E is measurable.
I believe that X_E(x) here is the characteristic function ( not sure about the name, but it is defined above).

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