Convergence in $L^p$ space

Question

Let $(X, M, \mu)$ be a $\sigma$-finite measure space.

Let $1<p < \infty$. Suppose $(f_n)_{n \in \mathbb{N}}$ is a bounded sequence in $L^{p}(\mu)$ such that $\lim_{n \rightarrow \infty} f_n g d\mu$ exists for every $g \in L^{q}(\mu)$.

Find $f \in L^p(\mu)$ so that $\lim_{n\rightarrow \infty}\int f_n g d\mu = \int fg d\mu$.

I thought that the last part requires the Lebesgue Dominated Convergence Theorem. But the assumptions don't meet the assumptions to apply the theorem here.

Any help will be appreciated!

Answer 1

If we allow the limit of $\int f_n gd\mu$ to depend on $g$, then the function $f$ could be any element of $L^p\pr{\mu}$ (take $f_n=f$).

If $\lim_{n\to\infty}\int f_n gd\mu$ is not allowed to depend on $g$, then (replacing $g$ by $-g$), one gets that $\lim_{n\to\infty}\int f_n gd\mu=0$ and only $f=0$ can work, since $\int fgd\mu=0$ for each $g\in L^q(\mu)$ implies that $g=0$.