Characterization of $L^p$ function

Question

Let $f:[0,1]\to\mathbb{R}$ be measurable and $p,q>1$ with $p^{-1}+q^{-1}=1$. Assume that $\int_{[0,1]} \left |f(t)g(t)\right |dt<\infty$ for all $g\in L^q([0,1])$. Does it follow that $f\in L^p([0,1])$. I have a hunch that the answer is no, but I haven't found a counter example. Thanks

Answer 1

Yes, it does. Let $$\eqalign{f_N(x) &= \cases{f(x) & if $|f(x)| \le N$\cr 0 & otherwise}\cr \phi_N(g) &= \int_\Omega f_N(x) g(x)\; dx\cr \phi(g)&= \int_\Omega f(x) g(x)\; dx\cr}$$ By Dominated Convergence, $\phi_N(g) \to \phi(g)$ as $N \to \infty$ for all $g \in L^q$. By the Uniform Boundedness Principle, there is $M$ such that all $\|\phi_N\| \le M$. This implies that $\phi$ is also a bounded linear functional on $L^q$, and then $f \in L^p$.