Representation of measures with polynomial denseness in $\mathbf{L}_{p}\, (\mathbb{R}, d\mu)$, $0<p<\infty$, and its application to determinate moment problems

It has been proved that algebraic polynomials $\mathcal{P}$ are dense in the space $L^{p}({\mathbb{R}},d\mu)$, $p\in(0, \infty)$, iff the measure $\mu$ is representable as $d\mu=w^p\, d\nu$ with a finite non-negative Borel measure $\nu$ and an upper semi-continuous function $w:\mathbb{R}\to\mathbb{R}^+:\,=[0,\infty)$ such that $\mathcal{P}$ is a dense subset of the space $C^0_w :\,= \{f\in C(\mathbb{R}) : w(x)f (x)\to 0 \,$   as$\, \vert x\vert\to\infty \}$ equipped with the seminorm $\Vert f \Vert _{w}:= \sup_{x \in{\mathbb{R}}} w(x)\vert f(x)\vert$. The similar representation $(1+x^2)d\mu=w^2 d\nu$ ( $(1+x)d\mu=w^2 d\nu$) with the same $\nu$ and $w$ ( $w(x)=0, x < 0$, and $\mathcal{P}$ is also a dense

subset of ${C^0_{\sqrt{x}\,\cdot\, w}}$) corresponds to all those measures (supported by $\mathbb{R}^+$) that are uniquely determined by their moments on $\mathbb{R}$ ( $\mathbb{R}^+$). The proof is based on de Branges' theorem (1959) on weighted polynomial approximation. A more general question on polynomial denseness in a separable Fréchet space in the sense of Banach $L^\Phi({\mathbb{R}},d\mu)$ has also been examined.