Sequence in $L^p$

Question

Let $1\leq p,q\leq \infty$ Hölder conjugates. Let $L^p:=L^p(\Omega,\mathcal{F},P)$, $L^q:=L^q(\Omega,\mathcal{F},P)$, where $(\Omega,\mathcal{F},P)$ is a probability space.

Let $(x_n)\subset L^p$ be a sequence of non-negative and disjoint variables (i.e. $\text{supp}(x_n)\cap \text{supp}(x_m)=\emptyset$ for all $n\neq m$) and so that $\Vert x_n\Vert_p=1$ for all $n\in\mathbb{N}$.

Is it true that we can find a sequence of non-negative and norm bounded random variables $(y_n)\subset L^q$ such that $\text{supp}(y_n)\subset \text{supp}(x_n)$, $\mathbb{E}[x_n y_n]=1$ for all $n\in\mathbb{N}$.

Answer 1

Assume that $1\lt p\lt \infty $. Define $y_n=x_n^{p-1}$. Then the support of $y_n$ is contained in that of $x_n$, $y_n$ is non-negative and $x_ny_n=x_n^p$ hence $\mathbb E\left[x_ny_n\right]=1$. Moreover, $y_n^q=x_n^p$ hence $\left(\left\lVert y_n\right\rVert_q\right)_{n\geqslant 1}$ is bounded.

If $p=1$, define $y_n:=\mathbf 1\left\{x_n\neq 0 \right\}$.