Real analysis - Lebesgue Integral
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For any positive integer k let (see attached) be the function from -> defined by (see attached). If (see attached) show that (show attached).
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 (see attached) 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: see attached.
Theorem 7.2: If f is Riemann integrable on [a,b], then it is Lebesgue integrable on [a,b] and (see attached)
Theorem 7.3 L is a linear space and the integral is a linear functional on L ; that is, if (see attached) L and (see attached), then (see attached) and (see attached) belong to L and (see attached) and (see attached).
Theorem 7.4 L is a lattice.
Theorem 7.5 If (see attached) L and (see attached) almost everywhere, then (see attached).
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