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Proof.
See the references mentioned above.
Theorem D.4.1(3) also holds in the unbounded case. The next example shows that,
unlike the Lebesgue integral, there are functions f with the property that f is Riemann
integrable but |f| is not.
D.4.2. Example.
The Riemann integral
sin x
x
•
Ú
dx
0
exists, but the Riemann integral
sin x
x
•
Ú
dx
0
diverges. See [Spie69].
Finally, the Riemann integral is only defined if the function is bounded. This is
not a requirement for the Lebesgue integral.
