Sufficient condition for “Weak Convergence”

This is a sufficient condition for something that resembles “Weak convergence”: \int f_kg\to \int fg for all g\in L^{p'}
Suppose that f_k\to f a.e.\ and that f_k, f\in L^p, 1<p\leq\infty. If \|f_k\|_p\leq M<\infty, we have \int f_kg\to\int fg for all g\in L^{p'}, 1/p+1/p'=1. Note that the result is false if p=1.

Proof:
(Case: |E|<\infty, where E is the domain of integration).

We may assume |E|>0, M>0, \|g\|_{p'}>0 otherwise the result is trivially true. Also, by Fatou’s Lemma, \displaystyle \|f\|_p\leq\liminf_{k\to\infty}\|f_k\|_p\leq M.

Let \epsilon>0. Since g\in L^{p'}, so g^{p'}\in L^1 and there exists \delta>0 such that for any measurable subset A\subseteq E with |A|<\delta, \int_A |g^{p'}|<\epsilon^{p'}.

Since f_k\to f a.e.\ (f is finite a.e.\ since f\in L^p), by Egorov’s Theorem there exists closed F\subseteq E such that |E\setminus F|<\delta and \{f_k\} converge uniformly to f on F. That is, there exists N(\epsilon) such that for k\geq N, |f_k(x)-f(x)|<\epsilon for all x\in F.

Then for k\geq N,
\begin{aligned}  \left|\int_E f_kg-fg\right|&\leq\int_E|f_k-f||g|\\  &=\int_{E\setminus F}|f_k-f||g|+\int_F|f_k-f||g|\\  &\leq\left(\int_{E\setminus F}|f_k-f|^p\right)^\frac{1}{p}\left(\int_{E\setminus F}|g|^{p'}\right)^\frac{1}{p'}+\epsilon\int_F |g|\\  &<\|f_k-f\|_p(\epsilon)+\epsilon\left(\int_F|g|^{p'}\right)^\frac{1}{p'}\left(\int_F |1|^p\right)^\frac{1}{p}\\  &\leq 2M\epsilon+\epsilon\|g\|_{p'}|E|^\frac{1}{p}\\  &=\epsilon(2M+\|g\|_{p'}|E|^\frac{1}{p}).  \end{aligned}

Since \epsilon>0 is arbitrary, this means \int_E f_g\to \int_E fg.

(Case: |E|=\infty). Error: See correction below.

Define E_N=E\cap B_N(0), where B_N(0) is the ball with radius N centered at the origin. Then |E_N|<\infty, so there exists N_1>0 such that for N\geq N_1, \int_{E_N}|f_k-f||g|<\epsilon.

Since |g|^{p'}\chi_{E_N}\nearrow|g|^{p'} on E, by Monotone Convergence Theorem, \displaystyle \lim_{N\to\infty}\int_{E_N}|g|^{p'}=\int_E |g|^{p'}<\infty.
Thus there exists N_2>0 such that for N\geq N_2, \int_{E\setminus E_N} |g|^{p'}<\epsilon^{p'}.

Then for N\geq\max\{N_1, N_2\},
\begin{aligned}  \int_E |f_kg-fg|&=\int_{E_N}|f_k-f||g|+\int_{E\setminus E_N}|f_k-f||g|\\  &<\epsilon+\left(\int_{E\setminus E_N}|f_k-f|^p\right)^\frac{1}{p}\left(\int_{E\setminus E_N}|g|^{p'}\right)^\frac{1}{p'}\\  &<\epsilon+\|f_k-f\|_p(\epsilon)\\  &\leq\epsilon+2M\epsilon\\  &=\epsilon(1+2M).  \end{aligned}
so that \int_E f_kg\to\int_E fg.

(Show that the result is false if p=1).

Let f_k:=k\chi_{[0,\frac 1k]}. Then f_k\to f a.e., where f\equiv 0. Note that \int_\mathbb{R} |f_k|=1, \int_\mathbb{R} |f|=0 so that f_k, f\in L^1(\mathbb{R}). Similarly, \|f_k\|_1\leq M=1.

However if g\equiv 1\in L^\infty, \int_\mathbb{R} f_kg=1 for all k but \int_\mathbb{R} fg=0.

Correction for the case |E|=\infty:

Define E_N=E\cap B_N(0), where B_N(0) is the ball with radius N centered at the origin.

Since |g|^{p'}\chi_{E_N}\nearrow |g|^{p'} on E, by Monotone Convergence Theorem, \displaystyle \lim_{N\to\infty}\int_{E_N}|g|^{p'}=\int_E|g|^{p'}<\infty.

Thus there exists N_1>0 such that \int_{E\setminus E_{N_1}}|g|^{p'}<\epsilon^{p'}.

Since |E_{N_1}|<\infty, by the finite measure case there exists N_2 such that for k\geq N_2, \displaystyle \int_{E_{N_1}}|f_k-f||g|<\epsilon.

So for k\geq N_2,
\begin{aligned}  \int_E|f_kg-fg|&=\int_{E_{N_1}}|f_k-f||g|+\int_{E\setminus E_{N_1}}|f_k-f||g|\\  &<\epsilon+\left(\int_{E\setminus E_{N_1}}|f_k-f|^p\right)^{1/p}\left(\int_{E\setminus E_{N_1}}|g|^{p'}\right)^{1/p'}\\  &<\epsilon+\|f_k-f\|_p(\epsilon)\\  &\leq\epsilon+2M\epsilon\\  &=\epsilon(1+2M).  \end{aligned}

so that \int_Ef_kg\to\int_E fg.

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