Show that there exitst $f \in L^{1}([0,1])$ such that $\int_{0}^{1}f(x)g(x)dx \nrightarrow 0$

Question

Define $$ g_{n} = n\mathbb{I}_{[0,\frac{1}{n^3}]}(x)\;\; $$ where $\mathbb{I}$ is index function. (if $x \in E, \mathbb{I}_{E}(x) = 1$, otherwise 0)

show that there exists $f \in L^{1}([0,1])$ such that $\int_{0}^{1}f(x)g_{n}(x)dx \nrightarrow 0$

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Could you give any idea to have counterexample for $f$?

Answer 1

Consider $f$ of the form $\sum_{n\geqslant 1}a_n\chi_{((n+1)^{-3},n^{-3})}$, where $a_n$ is a positive real number. Then $$f\in L^1\Leftrightarrow \sum_n\frac{a_n}{n^4}\lt\infty,\mbox{ and }$$ $$\int fg_n\mathrm dx\mbox{ behaves like }n\sum_{j\geqslant n}\frac{a_j}{j^4}.$$ So we have to choose the $a_j$ in order to make the series $\sum_n\frac{a_n}{n^4}$ convergent, but its remainder goes to $0$ slowly enough.