Actual procedure to do Lebesgue Integration

Question

Consider the following problem where one is asked to P.T.

$\int_1^\infty (\frac{1}{x}) dx = \infty$ as a Lebesgue integral.

How does one go about proving this?

Answer 1

$$ \left(\frac{1}{x} \chi([1,n])\right)_{ n \in \mathbb N}$$

where $\chi([1,n])$ is the characteristic function of the interval $[1,n]$, is an increasing sequence of non-negatives functions. So,

$$ \int_1^{\infty} \frac{dx}{x} = \int_{\mathbb R} \lim_{n \to \infty} \frac{\chi([1,n])dx}{x} = \lim_{n \to \infty} \int_{\mathbb R} \frac{\chi([1,n])dx}{x} = \lim_{n \to \infty} \ln (n) = \infty$$