Let $\mathbf{y}=\{Y_{n}\}_{n=1}^{\infty}$ be a sequence in aninfinite-dimensional Lebesgue space $L^{p}(\mathsf{\Omega},\mathcal{F},\mathrm{P})$, $p\geq1$, on some probability space $(\mathsf{\Omega},\mathcal{F},\mathrm{P})$. Let$$\mathsf{Y}=\left\{ \sum_{n=1}^{\infty}a_{n}Y_{n}:\mathbf{a}=\{a_{n}\}_{n=1}^{\infty}\in\ell^{\infty}\right\} \subset L^{p}$$be a proper subspace of $L^{p}$. Given $\mathbf{v}=\{v_{n}\}_{n=1}^{\infty}\in\ell^{1}$, there exists a positive continuous linear function$f:\mathsf{Y}\rightarrow\mathbb{R}$, such that $v_{n}=f(Y_{n})=\operatorname{E}(ZY_{n})$, with $Z>0$ and $Z\in(L^{p})^{\ast}$ (continuousdual space of $L^{p}$)
Show that we have an infinite number of positive linear functions such that$v_{n}=f_{X}(Y_{n})=\operatorname{E}(XY_{n})$ with $X>0$ and $X\in(L^{p})^{\ast}$.
If I can find a second one, then the convex combination will produce infinitely many of them. But how to find a second one? $\operatorname{E}(XY_{n})=\operatorname{E}(ZY_{n})$ with $Z\neq X>0$.