# Tagged Questions

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### Epi-convergence to indicator function

Let the indicator function be defined as $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and $I_{\nu}(x) \in [0,1]$ be a continuous approximation of ...
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### Why is semi lower continuity important for epi-convergence?

Why is the lower semicontinuity property important for epi-convergence (and, on the contrary, upper semicontinuity is not desirable)? A simple example would also help.
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### Product of strong and weakly converging sequences

Consider a sequence of functions $\{u_n\}\in L^2([0,T],L^2(\Omega))$ which converges strongly to a function $u\in L^2([0,T],L^2(\Omega))$. Then $u_n \rightarrow u \;\; a.e.$ in ...
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### Problem on $L^2$ spaces.

Let $f_n$ be sequence of continuous functions on $[0,1]$ converging uniformly to $f$ a.e. on a set of finite measure. I would like to prove that this implies $f_n\rightarrow f$ in the $L^2$ norm. ...
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### Why the convergence of the following operator consequence is strong?

Given a consequence of the following functional operators in $\mathcal{B}(L_{p}[0,1])$, $p \in [1,\infty)$ $$(A_{n}x)(t) = \sum_{k=1}^{n} n \int_{t_{k-1}}^{t_{k}} x(s)ds \chi_{k,n}(t),$$ where ...
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### Relation between convergences in $L^{p}$ for probability spaces.

I have read that for a probability space $(\Omega,\Sigma,P)$ it is true that $f \in L^{p}(\Omega,\Sigma,P)$ implies $f \in L^{q}(\Omega,\Sigma,P)$ if $p>q$, and hence $L^{2} \subset L^{1}$. I'm ...
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### Integral convergence and weak convergence

Given that $\Omega \subset \mathbb{R}^{n}$ is a connected bounded Lipshitz domain and $u_{k} \rightharpoonup u$ in $W^{1,p}(\Omega)$. We denote $\Gamma$ as the boundary of the domain. We have the ...
I know that for a bounded $\Gamma$ it follows that $L^{q}(\Gamma) \subset L^{p}(\Gamma)$ if $q > p$. I have a few questions regarding how $L^{p}$ spaces relate with regard to convergence. Consider ...