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I have a question about weak convergence and limit inferior.

Let $(X,\Sigma,\mu)$ be a measure space (if necessary $\sigma$-finite measure space). Let $(u_{t})_{t >0}$ be a family of square integrable functions (i.e. for every $t>0$, $u_{t} \in L^{2}(\mu)$) . Furthermore we suppose $\displaystyle \sup_{t>0}\|u_{t}\|_{L^{2}(\mu)}<\infty$ i.e. $(u_{t})_{t>0}$ is $L^{2}(\mu)$-bounded

Definition

$u \in L^{2}(\mu)$ is called a weak limit of $(u_{t})_{t>0}$ if there exists a sequence $(t_{n})_{n \in \mathbb{N}}$ converges to $0$ such that $u_{t_{n}} \underset{n \to \infty}\longrightarrow u$ weakly in $L^{2}(\mu)$

Question

Let $v$ is a square integrable function such that for every weak limit $u$ of $(u_{t})_{t>0}$, $v \leq u$ $\mu$-a.e. Then can we show that for all nonnegative square integrable function $\phi$, \begin{align*} (v,\phi)_{L^{2}(\mu)} \leq \varliminf_{t \to 0} (u_{t},\phi)_{L^{2}(\mu)} \end{align*} Where \begin{align*} \varliminf_{t \to 0} (u_{t},\phi)_{L^{2}(\mu)} =\sup_{\delta>0} \inf_{0<t<\delta} (u_{t},\phi)_{L^{2}(\mu)} \end{align*}

Attempt

Let $(t_{n})_{n \in \mathbb{N}}$ be a sequence which converges to $0$. Since $(u_{t_{n}})_{n \in \mathbb{N}}$ is $L^{2}(\mu)$ bounded, we can find subsequence of $(u_{t_{n}})_{n \in \mathbb{N}}$ and $u \in L^{2}(\mu)$ such that $u_{t_{n_{k}}} \underset{k \to \infty}\longrightarrow u$ weakly in $L^{2}(\mu)$. Therefore for all nonnegative $\phi \in L^{2}(\mu)$, \begin{align*} (v,\phi)_{L^{2}(\mu)} &\leq (u,\phi)_{L^{2}(\mu)} \\ &\leq \lim_{k \to \infty} (u_{t_{n_{k}}},\phi)_{L^{2}(\mu)} \\ &\left(= \varliminf_{k \to \infty} (u_{t_{n_{k}}},\phi)_{L^{2}(\mu)} \right) \end{align*} But $\displaystyle \varliminf_{t \to 0} (u_{t},\phi)_{L^{2}(\mu)} \leq \varliminf_{k \to \infty} (u_{t_{n_{k}}},\phi)_{L^{2}(\mu)} $ . Please tell me what I should do.

Thank you in agvance.

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You can do it the other way round. If $j_\phi = \varliminf_{t \to 0} (u_t, \phi)_{L^2}$, then you can find a sequence $\{t_n\}_{n \in \mathbb{N}}$ (depending on $\phi$), such that $j_\phi = \lim_{n \to \infty} (u_{t_n}, \phi)_{L^2}$.

This should help you to write down a proof.

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  • $\begingroup$ Thanks for your reply. I understood. Your explanation is very clear! $\endgroup$ – sharpe Jul 22 '15 at 6:29

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