I am given a function defined on $(\mathbb R,$ Borelians, Lebesgue measure) such that $$f= \begin{cases} + \infty &\text{if } x=0 \\ \ln(|x|) &\text{if } 0\lt |x|\lt1 \\ 0 &\text{if } |x|\ge1 \end{cases} $$

How can I show that this function is Lebesgue-integrable ?

  • $\begingroup$ Maybe you can use that the riemann integral of $f$ on $[-1, - 1/n] \cup [1/n, 1]$ for $n\geq 2$ is equal to the Lebesgue integral on the same interval. Construct a sequence of functions based on the previous observation and then try to apply monotone convergence theorem somewhere. What did you try btw? Do you know that the riemann integral is equal to the lebesgue integral? $\endgroup$ – Shashi Dec 1 '18 at 16:15

Since $f$ is even it is enough to show that it is integrable on $(0,1)$. But $$\int_{(0,1)} f(u)du =ulnu-u|_0^1 =1$$

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  • $\begingroup$ isn't there a way to show it using the definition of Lebesgue-integrability and basic theorems such as dominated convergence, monotone convergence? $\endgroup$ – guest Dec 1 '18 at 15:12

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