Let $X,Y\in L^1(\Omega,\mathcal F,\mathbb P)$, let $a,b\in\mathbb R$, and let $\mathcal G\subset\mathcal F$ be a sub-$\sigma$-algebra. Then
\begin{align*}
\mathbb E[aX+bY\mid\mathcal G]=a\mathbb E[X\mid\mathcal G]+b\mathbb E[Y\mid\mathcal G]
\end{align*}
$\mathbb P$-a.e. If $X\ge0$ $\mathbb P$-a.e., then
\begin{align*}
\mathbb E[X\mid\mathcal G]\ge0
\end{align*}
$\mathbb P$-a.e.