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Let $\mathcal M$ be the space of positive Radon measures $\nu$ on $\mathbb R^n$ with finite first order moment, bounded by a constant $M$ (i.e. $\int_{\mathbb R^n}d\nu\leq M$ for every $\nu\in\mathcal M$) and endowed with the weak star convergence topology. Such space is metrized by

$$ \text{d}_{\mathcal M}(\nu, \nu')=\sum_{h=1}^{\infty}\frac{1}{2^h}|\mathcal L_{\nu}(u_h)-\mathcal L_{\nu'}(u_h)|, $$

where

$$ \mathcal L_{\nu}(u)=\int_{\mathbb R^n}ud\nu,\quad u\in C^0_c(\mathbb R^n), \ \ \nu\in\mathcal M $$

and $(u_h)_h$ is a dense sequence in the unit ball of $(C_c^0(\mathbb R^n), \|\cdot\|_{\infty})$.

I know that in a compact subset $K\subset\mathbb R^n$, $\text{Lip(K)}$ is dense in $C^0_c(K)$ by the Stone-Weierstrass Theorem.

What I want to ask is: is it possible to approximate $u_h\in C^0_c(K)$ with a Lipschitz continuous function which has $h$ as Lipschitz constant? If not, what are the possible Lipschitz constants?

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    $\begingroup$ What does $\mathcal M$ has to do with your question? $\endgroup$
    – supinf
    Sep 23, 2020 at 14:20
  • $\begingroup$ Nothing. I wrote it only for the mathematical framework $\endgroup$
    – Redeldio
    Sep 23, 2020 at 14:28

1 Answer 1

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In general, this is not possible. Take $K=[0,1]$ and $u_h(x)=\sqrt{x}$ for a fixed $h$. Then we cannot approximate $u_h$ with functions that have $h$ as a Lipschitz constant, because every Limit of such functions would be Lipschitz continuous with Lipschitz constant $h$.

It is only possible to approximate $u_h(x)$ with growing Lipschitz constants.

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  • $\begingroup$ Ok, thank you. How can we estimate the growth of the Lipschitz constant? Do they grow less than $\frac{1}{2^h}$ as $h\rightarrow\infty$? $\endgroup$
    – Redeldio
    Sep 23, 2020 at 16:51
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    $\begingroup$ I do not know that. $\endgroup$
    – supinf
    Sep 23, 2020 at 20:40

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