Real analysis - Lebesgue Integral

(See attached file for full problem description)
For any positive integer k let   be the function from   defined by  .  If   show that  .



Section 7 Notes:  The Lebesgue Integral

Definition 7.1  Let L be the set of real-valued functions f such that for some g and h in   f=g-h almost everywhere.  The set L is called the set of Lebesgue integrable function on   and the Lebesgue integral of f is defined as follows:   .

Theorem 7.2:  If f is Riemann integrable on [a,b], then it is Lebesgue integrable on [a,b] and  

Theorem 7.3  L is a linear space and the integral is a linear functional on L ; that is, if   L and  , then  and   belong to L and   and  .  

Theorem 7.4   L is a lattice.

Theorem 7.5  If   L and  almost everywhere, then  .  

Attached file(s):

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