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Real Analysis : Measurable Sets and Functions

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).

Solution Preview

1. Proof:
Since f: X-->C is measurable, then for any open set A in C, f^(-1)(A) is a measurable set. We note that A=C-{0} is an open set, thus ...

Solution Summary

Measurable sets and functions are investigated.