How to explain $\delta(U-E) = \delta(U)- \delta(E)$?

Question

In proving the statement "$\delta E$ is Lebesgue measurable if $E$ is measurable" using the definition of Lebesgue measure, I face a problem about how to explain the set equation $\delta(U-E) = \delta U - \delta E$, where $\delta$ is a positive constant, and $\delta E :=\{\delta x: x \in E\}$. Can anyone explain it?

Answer 1

$f(x)=\delta x$ is a bijection of $\mathbb R$. Bijections preserve set operations: $f(A^{c})=(f(A))^{c}, f(\bigcup A_i)=\bigcup f(A_i), f(\bigcap A_i)=\bigcap f(A_i)$ etc. In particular $f(U-E)=fU)-f(E)$.