# Real analysis: Lebesgue Integral

Prove theorem 7.3 in notes attached.

Section 7: 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 .

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Section 7: 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 ...

#### Solution Summary

This solution is comprised of a detailed explanation to prove theorem 7.3 in notes attached.