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Let $f:[a,b]\to \mathbb R $ be an integrable function. Define $$f_+(x)=\begin{cases}f(x)&\text{if } f(x)\geq0,\\0&\text{if }f(x)<0, \end{cases} \text{ and } f_-(x)=\begin{cases}0 & \text{if }f(x)\geq0,\\-f(x) &\text{if }f(x)<0. \end{cases}$$ Prove that $f_+,f_-:[a,b]\to\mathbb{R}$ are integrable and $$\int_a^bf(x)dx=\int_a^bf_+(x)dx -\int_a^bf_-(x)dx.$$

I see that $f(x)=f_+(x)-f_-(x)$, so the last part seems fairly obvious. I'm not how to show the functions are integrable, though. I feel like I must be missing something obvious, but their upper and lower sums don't seem to relate nicely to $f(x)$.

Ainlaen
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  • Riemann or Lebesgue integrable? – Daniel McLaury Mar 17 '16 at 23:14
  • I guess it doesn't matter, it it's Riemann, http://math.stackexchange.com/questions/102844/how-do-i-prove-that-a-function-with-a-finite-number-of-discontinuities-is-rieman, then both $f_{+}$ and $f_{-}$ have a finite number of discontinuities (extra possible one at $x=0$). For Lebesgue - both $f_{+}$ and $f_{-}$ have a countable set of discontinuities (extra possible one at $x=0$). – rtybase Mar 17 '16 at 23:22
  • Riemann integrable. – Ainlaen Mar 17 '16 at 23:23
  • @Ainlaen It will be a lot easier to prove that they are Lebesgue integrable and from that point we can deduce that they are also riemann integrable. – christina_g Mar 17 '16 at 23:26
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    See http://math.stackexchange.com/q/306326/27978. The function $x \mapsto \max(0,x)$ is continuous. – copper.hat Mar 17 '16 at 23:28
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    @rtybase Actually I am pretty sure that for Lebesgue we can have an uncountable set of discontinuities so long as its lebesgue measure is 0. – christina_g Mar 17 '16 at 23:29
  • @christina_g: Yes, take $1_\mathbb{Q}$. – copper.hat Mar 17 '16 at 23:30
  • @christina_g, yes, you are right! – rtybase Mar 17 '16 at 23:32

1 Answers1

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Thanks to copper.hat's remark it finally made -clonk- and here it is: $$ \DeclareMathOperator{abs}{abs} f_+(x) = \max(f(x),0) = \frac{\abs(f(x))+f(x)}{2} \\ f_-(x) = -\min(f(x), 0) = \frac{\abs(f(x))-f(x)}{2} $$ We need the theorem that if $f$ is Riemann integrable on $[a,b]$, then so is $\abs(f)$ (Link). Together with linearity, this gives integrability of $f_+$ and $f_-$ on $[a,b]$.

Subtraction of the two equations above gives $f_+(x) - f_-(x) = f(x)$, together with the integrability and linearity it should yield the decomposition $$ \int\limits_a^b f(x) \, dx = \int\limits_a^b f_+(x) \, dx - \int\limits_a^b f_-(x) \, dx $$

mvw
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