# Lebesgue integral

• Jun 2nd 2010, 08:49 PM
willy0625
Lebesgue integral
I'm trying to evaluate

$\int_{-\infty}^{\infty}\frac{\lambda}{\vert x \vert}$

where $\lambda$ is the Lebesgue measure on $\mathbb{R}$ and the following is what I have done:

Let $f_{n}(x)=\frac{1}{\vert x \vert}\chi_{[-n,n]}.$ Then since $(f_{n}:n\in\mathbb{N})$ is non-negative and increasing, by the Monotone Convergnece Thorem we have

$\int_{-\infty}^{\infty}\frac{\lambda}{\vert x \vert}= \int_{\mathbb{R}}\lim_{n}\frac{1}{\vert x \vert}\chi_{[-n,n]}d\lambda = \lim_{n}\int_{\mathbb{R}}\frac{1}{\vert x \vert}\chi_{[-n,n]}\\=\lim_{n}\Big(\int_{(-n,0)}-\frac{1}{x}\lambda+\int_{(0,n)}\frac{1}{x}\lambda \Big)\hspace{1mm}$ $=\infty$,

since the integrals inside the limit notation coincides with Riemann integral.

Am I on the right track?
• Jun 7th 2010, 02:25 AM
drjerry
You're on the right track.
Jerry