TITLE: Is there an integrable function that dominates $f_n(x)=\frac{\chi_{[0,n]}(x)}{n \ln (n)}$?
QUESTION [0 upvotes]: Is there an integrable function (with respect to Lebesgue measure on $\Bbb R$) that dominates $\displaystyle f_n(x)=\frac{\chi_{[0,n]}(x)}{n \ln (n)}$ for all $n \geq 2$? How can I find it?

REPLY [0 votes]: If $g$ dominates this sequence then $\int_0^{\infty} g(x)dx\geq \sum_n \int_n^{n+1} g(x)dx \geq 
\sum_n \frac 1 {(n+1) \ln (n+1)}=\infty$. Hence no such dominating function exists.
{"set_name": "stack_exchange", "score": 0, "question_id": 3610407}