Deducing weak convergence of $f_n$ s.t for every $g \in L^q(\mu)$, $\lim_n \int f_n g \,d\mu$ converges

Question

Let $(X, \mathcal{A}, \mu)$ be a finite measure space. Fix $1 < p, q < \infty$ with $\frac{1}{p} + \frac{1}{q} = 1$. Let $\{f_n\}$ be a sequence in $\mathcal{L^p}(\mu)$ such that $\sup \|f_n\|_p = M < \infty$. Suppose for every $g \in \mathcal{L^q}(\mu)$ we have $\lim_n \int f_n g \, d\mu$ converges.

Prove that $f_n$ weakly converges to some $f \in L^p(\mu)$.

I have shown that $\phi(g) = \lim_n \int f_n g \, d\mu$ defines a linear functional $\phi \in (L^q(\mu))^*$. I think $f$ is the element in $L^p(\mu)$ corresponding to $\phi \in (L^q(\mu))^*$ by the Riezs' theorem isomorphism, but how do I prove this? How do I find this $f$?

Answer 1

If $\phi$ is a bounded linear functional on $L^q(\mu)$ then there exists $f \in L^p(\mu)$ with $\phi(g) = \displaystyle \int fg \, d\mu$. You don't have to find it; Riesz did that for you.

Under your definition of $\phi$ this means $$\lim_n \int f_ng \, d\mu = \int fg \, d\mu$$ for all $g \in L^q(\mu)$, verifying weak convergence.

Answer 2

Clearly, $\phi(g)=\lim \int_X f_ng\,d\mu$ is well-defined and linear. Moreover, $$|\phi(g)|\leq M\|g\|_q$$ so $\|\phi\|\leq M$.

By Riesz Representation, there exists a unique $f\in L^P$ such that $\phi(g)=\int_X fg\,d\mu$. But this means $\lim \int_X (f_n-f)g \,d\mu=0.$

Answer 3

Do you know the representation theorem for $L_p$? This one says that for $1\leq p < \infty$ each element of the dual space of $L_p$ can be written as $f\to \int fg \,d\mu$ for some $g\in L_q$.

Thus you now know that for any $F\in L_p'$ we have $F(f_n)$ converges. Now you only need to proof that it converges to $F(f)$ for a $f$ independent of $F$.

Note that the limit of $F(f_n)$ is linear in $F$. Thus $g\to\lim\int f_ng\,d\mu$ is itself linear, and clearly continuous by Cauchy-Schwarz and Fatou. Thus again by the representation theorem there exists a $f\in L^p$ so that this can be written as $\int fg \, d\mu$.

This implies $F(f_n)\to F(f)$ for each $F\in L_p'$, which means by definition weak convergence.