Take two classes of uniformly square integrable random variables $\lbrace X_t:t\in T\rbrace$ and $\lbrace Y_t:t\in T\rbrace$. Is the class $\lbrace X_tY_t :t\in T\rbrace$ uniformly integrable?
My guess is yes because if I choose any $\varepsilon >0$ and find $\delta _{X}>0$ and $\delta _{Y}>0$ and such that for all $A$ and $t$, $% \mu \left( A\right) <\delta _{X}\implies \int_{A}\left\vert X_{t}\right\vert ^{2}d\mu <\varepsilon $ and $\mu \left( A\right) <\delta _{Y}\implies \int_{A}\left\vert Y_{t}\right\vert ^{2}d\mu <\varepsilon $. I then take $\delta $ equal to the minimum of $\delta _{X}$ and $\delta _{Y}$. Then: for all $\mu \left( A\right) <\delta $ and $t$ \begin{eqnarray*} \int_{A}\left\vert X_{t}Y_{t}\right\vert d\mu &\leq &\sqrt{% \int_{A}\left\vert X_{t}\right\vert ^{2}d\mu \int_{A}\left\vert Y_{t}\right\vert ^{2}d\mu } \\ &<&\varepsilon \text{.} \end{eqnarray*}
If the assumption is that the families $(X_t^2)_{t\in T}$ and $(Y_t^2)_{t\in T}$ are uniformly integrable, then the result is true (we have to check that $(X_tY_t)_{t\in T}$ i s bounded in $\mathbb L^1$ which is not a hard task).
If $(X_t^2)_{t\in T}$ and $(Y_t^2)_{t\in T}$ are only bounded in $\mathbb L^1$, it does not work. Assume that $T=\mathbb N\setminus \{0\} $ and $X_t=Y_t=\sqrt t\mathbb 1_{(0,1/t) }$ on $(0,1)$ endowed with Lebesgue measure.
Then $\mathbb E(X_t^2)=1=\mathbb E(X_t^2)$ but $X_tY_t=t\mathbb 1_{(0,1/t) }$.