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The problem goes as follows: Let $f_n$ be strictly positive Lebesgue measurable function defined on $[0,\infty)$ satisfying $$\lim_{n \to \infty} \int_0^\infty f_n(x)\ dx=0$$ then show that there exists a positive, strictly increasing measurable function $b(\cdot)$ such that $\lim_{x \to \infty} b(x)=\infty$ and that $$\lim_{n \to \infty}\int_{0}^\infty b(x)f_n(x)\ dx =0$$

All that comes to me is another version:

Let $f$ be strictly positive Lebesgue measurable function defined on $[0,\infty)$ satisfying $$\int_0^\infty f(x)\ dx<\infty$$ then there exists a positive, strictly increasing measurable function $a(\cdot)$ such that $\lim_{x \to \infty} a(x)=\infty$ and that $$\int_{0}^\infty a(x)f(x)\ dx <\infty$$

I am not sure if this helps, but I initially hopes to find some function for each $f_n$ and then take $\inf$ for scaled ones, but it seems difficult to illustrate that thus constructed function will be strictly positive, so I am confused of what to do next.

Roy Han
  • 911
  • If you assume that $\sum_{n=1}^\infty \int_0^\infty f_n(x),dx < \infty$, then set $f(x)=\sum_{n=1}^\infty f_n(x)$ and apply the result that you mentioned. Can you see any way to reduce to this case? – Aaron Tikuisis Aug 18 '13 at 07:25
  • @AaronTikuisis I do not come up with the idea to reduce the initial condition to $\sum_{n\geq 1}\int_0^\infty f_n <\infty$...can you show me a little more details? – Roy Han Aug 19 '13 at 13:44

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