(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

Friedman defines "integrable" like this: An a.e. real-valued measurable function ##f:X\to\overline{\mathbb R}## is said to beintegrableif 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 L^{1}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.$$

3. 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: F is integrable if and only if its positive and negative parts are

**Physics Forums | Science Articles, Homework Help, Discussion**