# Real analysis : Measurable Functions

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Q1. If 0<=a_1<=a_2<=a_3<=....,( 1,2,3 are the subscripts of a)

0<=b_1<=b_2<=b_3<=......(1,2,3 are the subscripts of b)

and a_n --> a and b_n -->b

Then prove that a_n*b_n -->a*b

Q2.Let f: R --> R be monotonically increasing, i.e.

f(x_1)<= f(x_2) for x_1< = x_2.

Show that f is measurable.

Hint: You may extend f to f':[-infinity,infinity]-->[-infinity,infinity]and

show that (x_alpha,infinity] is contained in f'^-1((alpha,infinity])

is contained in [x_alpha,infinity],where x_alpha=inf{x:f'(x)>alpha}

Please respond with Complete proof/solution.

NOTE: f' means a bar on the head of f

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Measurable functions are investigated. The solution is detailed and well-presented.

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Question 1: Proof:

Since and , then for enough big , we have . This implies . In another word, when is big enough, is bounded by . ...

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