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Let $\{f_n\} \subseteq L^p,\{g_n\} \subseteq L^q, f \in L^p,g \in L^q,$ and ${1} \over {p}$ + ${1} \over {q}$$=1$.

Suppose $\|f_n-f\|_p \rightarrow 0$ and $\int_{\Omega} g_n \varphi d\mu \rightarrow \int_{\Omega} g \varphi d\mu$ for every $\varphi \in L^p$. Then is it true that $\int_{\Omega} f_n g_n d\mu \rightarrow \int_{\Omega} fg d\mu$ ?

$|\int_{\Omega}f_n g_n d\mu -\int_{\Omega} fg d \mu|=|\int_{\Omega}f_n (g_n-g) d\mu +\int_{\Omega} (f_n-f)g d \mu|\le \int_{\Omega} |f_n||g_n-g| d\mu+\int_{\Omega} |f_n-f| |g| d\mu$

$\int_{\Omega} |f_n-f| |g| d\mu\le ||f_n-f||_{L^p(\Omega)} ||g||_{L^q(\Omega)} \rightarrow 0$ (using Holder's inequality)

For the first integral : $\int_{\Omega} |f_n||g_n-g| d\mu$ in this case can I use weak convergence of g?

user3669039
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Giulia B.
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    what is $\varphi$? – Jake28 May 28 '20 at 13:17
  • I guess any $L^p$ function... By $\to$ do you mean $L^1$ convergence or just the convergence of the integrals as a sequence of numbers? – Mick May 28 '20 at 13:18
  • See proposition 3.5(iv) in Brezis' Functional Analysis, Sobolev spaces and Partial Differential Equations – Albert May 28 '20 at 13:19
  • You have this $\varphi$ in there with no explanation. – zhw. May 28 '20 at 15:56
  • my professor said that $f_j \rightarrow f$ in the $\tau_{X^}$ $X^-$topology if and only if $|\int f_j \varphi -\int f \varphi | \rightarrow 0, \forall \varphi \in C^{\infty}_c, f_j \in L^p$ – Giulia B. May 28 '20 at 16:05

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Observe that, for every $n \in \mathbb{N}$, we have $$ \begin{align} \left| \int f_n g_n - \int fg \right| &= \left| \int f_n g_n - \int f g_n + \int f g_n - \int f g \right| \\ &=\left| \int g_n (f_n -f ) + \int fg_n - \int fg \right| \\ &\leq \int |g_n||f_n-f| + \left| \int fg_n - \int fg\right|. \end{align} $$ Since $\{g_n\}_n$ converges weakly in $L^q$, it is bounded in $L^q$, namely there exists a constant $C>0$ such that $||g_n||_{L^q} \leq C \; \forall n \in \mathbb{N}$. By Holder's inequality, we have $$ \int |g_n||f_n-f| \leq ||g_n||_{L^q} ||f_n-f||_{L^p} \leq C ||f_n-f||_{L^p} \to 0 \text{ as } n \to \infty. $$ Since $f\in L^p$, by assumption we have $$ \left| \int fg_n - \int fg\right| \to 0 \text{ as } n \to \infty $$ and this concludes the proof.

user3669039
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