Let $\mu$ be a $\sigma$-finite translation invariant measure defined on the Borel subsets of $\mathbb R^d$ and $\lambda$ be the usual Lebesge measure. My question is how the Fubini theorem is used in the following argument I read and whether it is correct to use Fubini theorem in the first place. Thank you!
Let $A \subset \mathbb R^d$ be a Borel set with $\lambda(A)=0$. Define $\tilde A:= \{(x, y) \in \mathbb R^d \times\mathbb R^d: x+y \in A\}$. Hence, the $x$-slice $\tilde A_x$ of $\tilde A$ is $\{y\in\mathbb R^d: x+y \in A \}$ and the $y$-slice $\tilde A_y$ of $\tilde A$ is $\{x\in\mathbb R^d: x+y \in A \}$. Hence, by Fubini one has $(\mu \times \lambda)(\tilde A)=0$. And hence $\mu (B-x)$ for $x$ a.e.
$\underset{R^{d}\times R^{d}}{\iint}\mathbf{1}_{\overline{A}}\left(x,y\right)d\mu\times\lambda=\underset{R^{d}}{\intop}\left[\underset{R^{d}}{\int}\mathbf{1}_{\overline{A}}\left(x,y\right)d\mu\right]d\lambda=\underset{R^{d}}{\intop}\mu\left(\overline{A}_{y}\right)d\lambda=0 $
You may always use fubini when the function is positive and measurable.