Sequences in $L^p$ and Orlicz spaces Let $(\Omega,\mathcal{F},P)$ be a probability space. 
Let $1\leq p,q<\infty$ be Hölder conjugates. 
Let $L^p:=L^p(\Omega,\mathcal{F},P)$, $L^q:=L^q(\Omega,\mathcal{F},P)$.
Suppose that $(x_n)\subset L^p$ is a sequence of non-negative and disjoint random variables (i.e. $\text{supp}(x_n)\cap\text{supp}(x_m)=\emptyset$ for all $n\neq m$), and such that $\Vert x_n\Vert_p=1$ for all $n\in\mathbb{N}$. 
If $p>1$, set $y_n:=x^{p-1}_n$ for each $n$. Then $(y_n)\subset L^q$ is a norm bounded sequence of non-negative random variables with  $\mathbb{E}[x_n y_n]=1$ and $\text{supp}(y_n)\subset\text{supp}(x_n)$ for all $n\in\mathbb{N}$.
If $p=1$, the sequence $(y_n)$ with $y_n:=1_{\{x_n>0\}}$ is a sequence in $L^\infty$ which enjoys the same properties.
Suppose now that $\varphi:[0,\infty)\rightarrow[0,\infty)$ is a Young function and  $\psi$ is its conjugate Young function. Suppose also that $(x_n)\subset L^\varphi$ is a sequence of non-negative and disjoint random variables and such that $\Vert x_n\Vert_\varphi=1$ for all $n\in\mathbb{N}$. 
My question is: Can we find a norm bounded sequence $(y_n)\subset L^\psi$ of non-negative random variables with  $\mathbb{E}[x_n y_n]=1$ and $\text{supp}(y_n)\subset\text{supp}(x_n)$ for all $n\in\mathbb{N}$. 
 A: I don't know much about this stuff, but let's give it a try. I use the definitions from the Wikipedia page on Birnbaum-Orlicz spaces. We have
$$
1 = \|x_n\|_\varphi = \sup\left\{\int_\Omega x_ny\,d\mu : y\ge 0\text{ measurable and }\int_\Omega\psi\circ y\,d\mu\le 1\right\}.
$$
Hence, for each $n$ there exists some measurable $y_n'\ge 0$ such that $\int_\Omega\psi\circ y_n'\,d\mu\le 1$ and $a_n := \mathbb E[x_ny_n'] = \int_\Omega x_ny_n'\,d\mu\ge 1-\tfrac 1 n$. Now, set
$$
y_n := a_n^{-1}\cdot 1_{x_n > 0}\cdot y_n',
$$
where $1_\Delta$ denotes the characteristic function w.r.t. the set $\Delta$. Surely, with this definition we have $\operatorname{supp}y_n\subset \operatorname{supp}x_n$. Also,
$$
\mathbb E[x_ny_n] = \int_\Omega x_ny_n\,d\mu = a_n^{-1}\int_\Omega x_ny_n'\,d\mu = 1.
$$
It only remains to show that $y_n\in L^\psi$ and that $(y_n)$ is norm-bounded. For this, let us define $y_n'' := 1_{x_n > 0}\cdot y_n'$. Then, as $\psi(0) = 0$ and $\psi\ge 0$, we have
$$
\int_\Omega\psi\circ y_n''\,d\mu = \int_{x_n>0}\psi\circ y_n'\,d\mu\,\le\, \int_\Omega\psi\circ y_n'\,d\mu\,\le\,1.
$$
Using the Rao-Ren norm (from the Wiki article), this implies $y_n''\in L^\psi$ and $\|y_n''\|_\psi'\le 1$. Hence, $\|y_n\|_\psi' = a_n^{-1}\|y_n''\|_\psi'\le a_n^{-1}\le 2$. So, $(y_n)$ is bounded in the Rao-Ren norm $\|\cdot\|_\psi'$, which is equivalent to $\|\cdot\|_\psi$. Thus, $(y_n)$ is norm-bounded.
