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Prove Functions and Sets

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1)
a. Let f be defined on [a, b] by f(x) prove directly that f is measurable
b. let E be measurable subset of R, and let f be measurable function on E Define the function f and f on E as follows: f (x) = max { f(x), 0}, and f (x) = max{ -f(x), 0},

1b. prove directly that f and f are nonnegative measurable function on E with f(x) = f (x) - f (x).

2b. prove that = and that is measurable.

3b. if f is areal- valued function on [a, b] such that is measurable on [a, b] is f measurable on [a,b]?

4b. if f(x)= + cos(x), x [0, 2 ], find f (x) and f (x).
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This solution is comprised of a detailed explanation to prove functions.

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