\begin{thm}[\B{Fubini's Theorem}]
Let $A\subset\Real^n$ and $B\subset\Real^$ be closed rectangles, and let $f: A \times B \to \Real$ be integrable. For $x\in A$ let $g_x : B\to\Real$ be defined by $g_x(y) = f(x,y)$ and let
\begin{equation}
\mathfrak{L} (x) := \LL \int_B g_x = \LL \int_B f(x,y) dy,
\end{equation}
\begin{equation}
\mathfrak{U} (x) := \UU \int_B g_x = \UU \int_B f(x,y) dy.
\end{equation}
Then $\mathfrak{L}$ and $\mathfrak{U}$ are integrable on $A$ and
\begin{equation}
\int_{A\times B} f = \int_A \mathfrak{L} = \int_A \left( \LL \int_B f(x,y)dy \right) dx,
\end{equation}
\begin{equation}
\int_{A\times B} f = \int_A \mathfrak{U} = \int_A \left( \UU \int_B f(x,y)dy \right) dx.
end{equation}
\end{thm}