\begin{tikzpicture}[fill=gray]
% left hand
\scope
\clip (-2,-2) rectangle (2,2)
(1,0) circle (1);
\fill (0,0) circle (1);
\endscope
% right hand
\scope
\clip (-2,-2) rectangle (2,2)
(0,0) circle (1);
\fill (1,0) circle (1);
\endscope
% outline
\draw (0,0) circle (1) (0,1) node [text=black,above](A) {$L(A)$}
(1,0) circle (1) (1,1) node [text=black,above](B) {$L(B)$}
(-2,-2) rectangle (3,2) node [text=black,above] {$H$};
\node[below] at (barycentric cs:A=1/2,B=1/2 ) {$A \cap B$};%I don't know how to orient here
\end{tikzpicture}
\end{center}
I want the intersecting portion to be labeled as