Integrating function $h:=\lambda(A \cap (B-x))$ for $\lambda(A),\lambda(B)<\infty$

Question

Given two sets $A,B$ with finite measure let $h:=\lambda(A \cap (B-x))$. Show this function is integrable and calculate its integral.

I thought about using the following identity, $\lambda(A)+\lambda(B)=\lambda(A \cup B)+\lambda(A \cap B)$, approximating both $A$ and $B$ from below with intervals, and using the monotone convergence theorem.

Thanks in advance for any help.

Answer 1

$\lambda (A\cap (B-x)=\int 1_{A}(y)1_B(x+y)dy$

Use Tonelli's theorem to show that $\int_{\Bbb{R}} \int_{\Bbb{R}}F(x,y)dydx<\infty$ where $F(x,y)=1_{A}(y)1_B(x+y)$ is non-negative.

The integral is equal to $m(A)m(B)$ by translation invarianve of the Lebesgue measure.