A characterization of Weak Convergence in $L^p$ spaces

Question

I'm working on the following problem, I'm having trouble with the reverse direction. My question is bolded below. Also could someone check my forward direction?:

Let $(X, \mathcal{M}, \mu)$ be a $\sigma$ finite measure space and $\{f_n\},f \in L^P(X)$. Prove that $f_n \rightharpoonup f$ in $L^p(X)$ iff $\|f_n\|_p \leq c$ for all $n$ and $\int_A f_n\, d\mu \rightarrow \int_A f \, d\mu$ for all $A$ with $\mu(A) < \infty$.

For the reverse direction, we can use the characteristic functions in $L^q$ to build arbitrary functions in $L^q$ and use Monotone Convergence on $A$ equals a ball. Then increase the radius of the ball at each step making error $\epsilon/2^n$. However, I'm having trouble seeing how I use the boundedness of the sequence $f_n$)

(For the forward direction, choosing $\chi_{A}\in L^q(X)$ will get the integral condition and the $\|f_n\|_p$ were bounded because the sequence originally lived in $L^p(X)$.)

Answer 1

Here is how the boundedness of the sequence $\{f_n\}$ should enter your argument in the reverse direction:

Let $g\in L^q$. You want to show that $\int f_n g\, d\mu \rightarrow \int f g \,d\mu$. To this end, construct a sequence of simple functions $g_m$ such that $g_m\rightarrow g$ strongly in $L^q$ (I guess this is what you mean by "build"). You can then deduce

$$ \begin{split} \left|\int (f_n-f) g\, d\mu \right| &\leq \left|\int (f_n-f) (g-g_m)\, d\mu \right| + \left|\int (f_n-f) g_m\, d\mu\right|\\ &\leq \|f_n-f\|_p\|g-g_m\|_q + \left|\int (f_n-f) g_m\, d\mu\right| \\ &\leq (\|f_n\|_p+\|f\|_p)\|g-g_m\|_q + \left|\int (f_n-f) g_m\, d\mu\right| \\ &\leq (C+\|f\|_p)\|g-g_m\|_q+ \left|\int (f_n-f) g_m\, d\mu\right| <\epsilon \end{split} $$

for $n$ sufficiently large, provided $\|f_n\|_p$ is bounded by $C$. More precisely, you first choose $m$ sufficiently large so that the first term above is less than, say, $\frac{\epsilon}{2}$, then choose $n$ sufficiently large so that the second term above is less than $\frac{\epsilon}{2}$.

P.S. Your should include your assumptions on $p$. I am assuming $p\in(1,\infty)$.

Answer 2

This answer will only deal with the forward implication since you have a good answer dealing with the other one.

First, let's show that weak convergence implies boundedness of norms. This is in fact true in Banach spaces in general by essentially the same idea but I will work in this special case.

Recall that $L^p(X)^*$ is isometrically isomorphic to $L^q(X)$ where $p^{-1} + q^{-1} = 1$. In particular, we can identify each $f_n$ with a linear functional $\phi_n$ on $L^q$ defined by $$\phi_n(g) = \int_X f_n g d\mu$$ and have $\|f_n\| = \|\phi_n\|$.

Now we seek to apply the uniform boundedness theorem to the family $\{\phi_n\}_{n \geq 1}$. To do this, notice that for fixed $g \in L^q(X)$, $\phi_n(g) \to \int_X fg d\mu$ by weak convergence of $f_n$. Since convergent sequences in $\mathbb{R}$ (or $\mathbb{C}$) are bounded this implies that $|\phi_n(g)|$ is a bounded sequence for each $g \in L^q(X)$.

In turn, by the Uniform Boundedness Theorem, we have that $\sup_n \|f_n\| = \sup_n \|\phi_n\| < \infty$.

The second part is immediate since $1_A \in L^q(X)$ so that $$\int_A f_n d \mu = \int_X 1_A f_n d \mu \to \int_X 1_A f d \mu = \int_A f d \mu$$ by the weak convergence.

Answer 3

For the forward direction:

Define $T_n(g):=\int_X f_n g$ to be a family of linear operators on $L^q$. Also, define $T(g):=\int_X f g$. Then for each $g,$

$$ |(T_n-T)(g)|\leq ||f_n-f||_p ||g||_q<C(g). $$ Note this bound depends on $g,$ but not $n$ because $f_n$ converges to $f$ in $L^p.$ By the uniform boundedness theorem, the dependence on $g$ can go away and so $||T_n-T|| \to 0$ in operator norm. By duality, $||T||=||f||_p$ and $||T_n||=||f_n||_p.$ Therefore, for large $n,$ we have $||T_n||<||T||+\epsilon$. This is equivalent to $||f_n||_p < ||f||_p + \epsilon.$ The second part is trivial by Holder's inequality.