lebesgue integral problem 4

liyi

$f\in L(\mathbb{R})$ and $x f(x) \in L(\mathbb{R})$ . Let
$F(x) = \int_{-\infty}^x f(t)dt$ .
If $\int_{-\infty}^\infty f(x)dx = 0$ show that $F\in L(\mathbb{R})$ .

**Posted:** Sun Dec 26, 2004 7:07 pm

Kent Merryfield

Note that $F(x)$ can also be written as $-\int_x^{\infty}f(t)dt.$

Define $G(x)=\int_{-\infty}^x|f(t)|dt$ for $x<0$ and $G(x)=\int_x^{\infty}|f(t)|dt$ for $x>0$ . Clearly, $|F(x)|\le G(x)$ so that if we prove $G$ integrable, we also prove that $F$ is integrable.

Note that for $x>0$ , $xG(x)=x\int_x^{\infty}|f(t)|dt\le \int_x^{\infty}t|f(t)|dt.$ Similarly, for $x<0$ , $|x|G(x)\le \int_{-\infty}^x|tf(t)|dt.$

Hence, $\lim_{|x|\to\infty}|x|G(x)=0.$

Integrate by parts (the justification for this is actually Fubini's Theorem):

$\int_{-\infty}^0G(x)dx=\left xG(x)\right|_{-\infty}^0 -\int_{-\infty}^0x|f(x)|dx,$ which is clearly finite because of the limit we established above.

A similar integration by parts argument shows the integrability of $G$ on $[0,\infty).$

**Posted:** Mon Dec 27, 2004 1:31 am

dickchimney · Poincare Conjecture Offline Joined: 06 Mar 2004 Posts: 218

Somone comments on the Kent's proof??!!
I think this problem can be consider Solved Wink

**Posted:** Thu Dec 30, 2004 12:08 am

Myth

I am waiting liyi says his word. Smile

**Posted:** Thu Dec 30, 2004 1:40 am

_________________
Myth is out of here

liyi

his proof is completely correct.
anyway, i was trying to avoid integration by parts.

**Posted:** Fri Jan 14, 2005 5:58 am