I have the following exercise: Let $f$ a real-valued Lebesgue measurable function on $[0,1 ]$, if $fg$ is Lebesgue integrable on $[0,1]$ for any function $g\in L^p([0,1],\mathbb{R})$ with $p \in [1,\infty) $, then $f\in L^{\frac{p}{p-1}}([0,1],\mathbb{R})$.
This is an exercise I remember from my Lebesgue Measure course, I think we proved the results for simple functions and then extend the result to $f$. But here I need to use the uniform boundedness principle, so I think I somehow need a sequence of operators, but I really don't know how to do this.
Any hints appreciated!
Hints: Define $T_n(f)=\int fg\chi_{\{|f| \leq n\}}$. You have to verify that $\|T_n\|=(\int |f|^{q} \chi_{\{|f| \leq n\}})^{1/q}$ where $q =\frac p {p-1}$. For this choose $g =sgn(f)|f|^{1/(p-1)}$. Once you get $\|T_n\|$ the result follows immediately from Uniform Boundedness Principle.