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Let $1<p<\infty$. Let $\{f_k\}$ be a sequence in $L^p(\mathbb{R}^n)$. Suppose $f_k\to f$ a.e. and there exists $C>0$ such that $||f_k||_p\leq C$ for all $k$. Prove that for all $g\in L^q(\mathbb{R}^n)$, $\int f_kgd\mu\to\int fgd\mu$, where $1/p+1/q=1$.

How to solve this? Thanks.

I have tried but failed: Let $\epsilon>0$. Since $||g||_q<\infty$, there exists a compact subset $K$ of $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n\setminus K} |g|^qd\mu <\epsilon$. By Egoroff theorem, since $\mu(K)<\infty$, there exists $F\subseteq K$ such that $\mu(F)<\epsilon$, $f_k\to f$ uniformly on $K\setminus F$. Hence $$ \int_{K\setminus F}|(f_k-f)g|d\mu\leq (\int_{K\setminus F}|f_k-f|^pd\mu)^{1/p}||g||_q\to 0. $$ Hence by $$ \int_{\mathbb{R}^n}|(f_k-f)g|d\mu=\int_{K\setminus F}|(f_k-f)g|d\mu+\int_F|(f_k-f)g|d\mu+\int_{\mathbb{R}^n\setminus K}|(f_k-f)g|d\mu, $$ we need to prove $$ \int_F|(f_k-f)g|d\mu\to 0 $$ and $$ \int_{\mathbb{R}^n\setminus K}|(f_k-f)g|d\mu\to 0. $$ Note that the followings may happen

(1). $\int |f|^p=\infty$;

(2). $\{f_k\}$ is not dominated by an integrable function.

Hence I cannot continue...

Shiquan
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    Have a look here: http://math.stackexchange.com/questions/442550/pointwise-convergence-and-boundedness-in-norm-imply-weak-convergence – PhoemueX Jun 14 '14 at 07:33

1 Answers1

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Using Fatou's lemma, $$\int |f|^pd\mu\leqslant \liminf_{n\to\infty}\int|f_n|^pd\mu\leqslant C,$$ hence case (1) cannot happen.

Using Hölder's inequality, we get $$\int_F|f_n-f||g|d\mu\leqslant \lVert f_n-f\rVert_p\left(\int_F|g|^qd\mu\right)^{1/q}\leqslant 2C\left(\int_F|g|^qd\mu\right)^{1/q}.$$ We can conclude using the following fact:

If $h$ is integrable and $\varepsilon\gt 0$, there exists $\delta>0$ such that for any $S$ of measure smaller than $\delta$, $\int_S|h|\lt\varepsilon$.

(hence there is a minor modification of the argument: we fix $g$, $\varepsilon$, the corresponding $\delta$ and Egoroff's theorem with this $\delta$ and not $\varepsilon$.

Davide Giraudo
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