Are there any results for lebesgue measurable sets that shows that I can decompose a set $A \subseteq R^d$ with infinite measure into pariswise disjoint sets of finite measure?
I know for example that if $A$ is measurable then the difference of $A$ and some $F_\sigma$ set has measure zero, but I cannot find any results about disjointness of the decomposition.
Well, yes: write $\Bbb R^d$ as the disjoint union of differences of open balls $$\Bbb R^d=\bigcup_{n\ge 1} B(0,n)\setminus B(0,n-1)$$ and therefore $A=\bigcup\limits_{n\ge 1} A_n$ with $A_n=A\cap B(0,n)\setminus B(0,n-1)$.