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Prove that $f(x)=x$ is not Lebesgue integrable on $[1, \infty)$. (hint: use def. of Lebesgue integrability for positive functions).

hint: Use integral $= \infty$ by defining simple function and $\chi_A$ characteristic function.

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1 Answer 1

It is hard to give a rigorous proof if we don't know the theorems you already know concerning Lebesgue Integrals, nor your definitions.

A very easy way to see it is if you know that the Lebesgue Integral is monotone:

$f(x)=x \geq 1$ on $[1,\infty)$, therefore

$$\int_{[1,\infty)} f(x) d\mathcal{L}(\mathbb{R})\geq\int_{[1,\infty)} 1 d\mathcal{L}(\mathbb{R})=\mathcal{L}([1,\infty))=\infty$$

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