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Homework Statement
Problem 2.6.3. in "Foundations of modern analysis", by Avner Friedman. Let f be a measurable function. Prove that f is integrable if and only if f+ and f- are integrable, or if and only if |f| is integrable.
Homework Equations
Friedman defines "integrable" like this: An a.e. real-valued measurable function ##f:X\to\overline{\mathbb R}## is said to be integrable if there's a sequence ##\langle f_n\rangle## of integrable simple functions that's Cauchy in the mean and such that ##f_n\to f## a.e.
"Cauchy in the mean" means that it's a Cauchy sequence with respect to the L1 norm (which hasn't been defined at this point). In other words, ##\langle f_n\rangle## is Cauchy in the mean if for all ##\varepsilon>0## there's an ##N\in\mathbb Z^+## such that for all ##n,m\in\mathbb Z^+##,
$$\newcommand{\dmu}{\ \mathrm{d}\mu}
n,m\geq N\ \Rightarrow\ \int|f_n-f_m|\dmu<\varepsilon.$$
The Attempt at a Solution
I haven't looked at the "Cauchyness" yet. I'm still just focusing on the part about convergence a.e. If ##f^+, f^-## are integrable, it's easy to show that f is, because there are sequences ##\langle (f^\pm)_n\rangle## that converge a.e. to ##f^\pm##, and we just need to find another sequence ##\langle f_n\rangle## that converges a.e. to f. All we have to do is to define ##f_n=(f^+)_n-(f^-)_n##, and the result ##f_n\to f## a.e. follows from
$$|f_n-f|=|(f^+)_n-(f^-)_n-f^+-f^-|\leq|(f^+)_n-f^+|+|(f^-)_n-f^-|.$$ The proof that |f| is integrable is very similar. It's the converses of these results that I'm struggling with. In particular, how do I prove that if f is integrable, then its positive and negative parts are integrable too?
It actually looks impossible to me. At least if I start with the "obvious" idea. Let ##\langle f_n\rangle## be a sequence of integrable simple functions that's Cauchy in the mean and such that ##f_n\to f## a.e. Then break each ##f_n## into positive and negative parts, and define the sequences ##\langle (f^\pm)_n\rangle## by ##(f^\pm)_n=(f_n)^\pm##. (I'll just write ##f^\pm_n## from now on).
Consider an x such that f(x)=0. We have ##f^\pm(x)=0##, but I don't see a reason why we can't have something like ##f^\pm_n\to\pm 1##.
On the other hand, if I assume that both f and |f| are integrable (instead of just f), I can prove that f+ and f- are integrable with a simple triangle inequality argument.