The $T$-strong duals of $L^1(T)$ and $L^{\mathrm{\infty }}(T)$

For a conditional expectation operator $T$ on a Dedekind complete Riesz space, we give representations of the $T$-strong duals of $L^1(T)$ and $L^{\mathrm{\infty }}(T)$. The representation for the $T$-strong dual of $L^1(T)$ follows from the known result for $L^2(T)$. To describe the $T$-strong dual of $L^{\mathrm{\infty }}(T)$, we introduce charges on components of weak order units and develop a corresponding integration theory.