How should I calculate the Lebesgue integral of logarithm function from zero to infinity?

Question

Does the area under the $\ln(x)$ in $(0,+\infty)$ is measurable? If yes, how can I calculate it?

Answer 1

The area between $0$ and $1$ is the negative of $\int_{-\infty}^0 e^x\;dx$ which is finite. On the other hand, the area between $1$ and $\infty$ is infinite (for example $\log x$ is larger than the indicator function of $(e,\infty)$.

Answer 2

For $x\geq e$, you have $\ln x\geq \ln e=1$. So the area under the curve on $[e,+\infty)$ is infinite.

Answer 3

$$\int\limits_0^\infty\log x\,dx=\lim_{b\to\infty,\,\epsilon\to 0}\int\limits_\epsilon^b\log x\,dx=\left.\lim_{b\to\infty,\,\epsilon\to 0}\left(x\log x-x\right)\right|_\epsilon^b=$$

$$=\lim_{b\to\infty,\,\epsilon\to 0}\left[\left(b\log b-b\right)-\left(\epsilon \log\epsilon-\epsilon\right)\right]=\infty$$