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Royden Chapter 4, Problem 10a

Show that if f is integrable over E then so is |f| and [tex]\left|\int_E f \right| \leq \int |f|[/tex].

Does the integrability of |f| => the integrability of f?

2. Relevant equations

[tex]f^+ = max\{f, 0\}[/tex]

[tex]f^- = max\{-f, 0\}[/tex]

[tex]|f| = f^+ + f^-[/tex]

A function is integrable if [tex]\int f < \infty[/tex]

3. The attempt at a solution

I have [tex]\left|\int_E f \right| \leq \int |f|[/tex]

[tex]\left|\int_E f\right| = \left| \int_E f^+ - \int_E f^- \right| \leq \left| \int_E f^+ \right| + \left| \int_E f^- \right| = \int_E |f|[/tex].

Next I want to show that if f is integrable over E then so is |f|.

[tex]\int_E f < \infty [/tex]

[tex]\int_E f^+ - \int_E f^- < \infty [/tex]

f+ and f- are finite because the expression a-b is only finite if both a and b are finite.

[tex]\int_E f^+ < \infty [/tex]

[tex]\int_E f^- < \infty [/tex]

Hence:

[tex]\int_E |f| = \int_E f^+ + \int_E f^- < \infty [/tex].

Does the integrability of |f| => the integrability of f?

I would say yes for similar reasons. I don't feel very confident about my answers here. Is the reasoning correct?

Also part b says that the improper Riemann integral (a limit) may exists for a function when the Legesgue integral fails to exists. and gives (sin x)/x as an example. Is the problem with (sin x)/x that f+ and f- are not finite??

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# Homework Help: If f is integrable over E iff |f| is integrable over E.

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