# Possible application of Stone-Weierstrass Theorem

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?

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

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.

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