- #1

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in a solution to my measure theory assignment, I have seen the equation

$$

\int_{\mathbb{R}}^{} \frac {1}{|x|}\, d\lambda(x)=\infty

$$

with ##\lambda## as the 1⁻dim Lebesgue measure.

I was wondering how that integral was evaluated as we had never proven any theorem that states that Lebesgue integrals can be computed as improper Riemann integrals. This simple equation makes it look trivial, but I don't see the reasoning. Under which condition can I just rewrite the integral as ## \int_{-\infty}^{\infty} \frac {1}{|x|}\, dx ##?