# A really strange question here in my opinion: Let A , B be two measurable subsets of

A really strange question here in my opinion:
Let $A,B$ be two measurable subsets of ${\mathbb{R}}^{1}$. Define $f\left(x\right)=|\left(A-x\right)\cap B|$. Evaluate ${\int }_{{\mathbb{R}}^{1}}fdx$. Here $|\cdot |$ refers to the measure.
Here $f$ is clearly non-negative, so I tried using the definition of Lebesgue integral:
${\int }_{{\mathbb{R}}^{1}}fdx=sup\left\{{\int }_{{\mathbb{R}}^{1}}sdx:0\le s\le f\right\},$
where s are simple functions. But I realised that I couldn't come up with any simple functions.
I tried breaking up ${\mathbb{R}}^{1}$ into two parts:
${E}_{1}=\left\{y\in {\mathbb{R}}^{1}:y\in A\cap B\right\},{E}_{2}=\left\{y\in {\mathbb{R}}^{1}:y\notin A\cap B\right\}$
and then considering the sum
${\int }_{{E}_{1}}fdx+{\int }_{{E}_{2}}fdx$
and then I'm clueless as to how to proceed. I'm guessing the answer should be something intuitive like $|A\cap B|$. Any help would be appreciated.
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Buckokasg
One could proceed as follows. Note that we can rewrite $f\left(x\right)=|\left(A-x\right)\cap B|={\int }_{\mathbb{R}}{1}_{B}\left(y\right){1}_{A}\left(x+y\right)dy$. Then the question becomes evaluating the expression
${\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{1}_{B}\left(y\right){1}_{A}\left(x+y\right)dydx.$
By translation invariance of the Lesbeque measure on $\mathbb{R}$, it follows that
${\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{1}_{B}\left(y\right){1}_{A}\left(x+y\right)dydx={\int }_{\mathbb{R}}{\int }_{\mathbb{R}}{1}_{B}\left(y\right){1}_{A}\left(y\right)dydx={\int }_{\mathbb{R}}{1}_{B}\left(y\right)dy{\int }_{\mathbb{R}}{1}_{A}\left(x\right)dx=|A|\cdot |B|.$