Consider the measure space $(\mathbb R, \mathcal B(\mathbb R), \lambda)$, where $\lambda$ denote the Lebesgue-measure on $\mathbb R$. Let $B \in \mathcal B(\mathbb R)$ and $f:(0, \infty) \rightarrow [0, \infty)$ be given by $f(x) := \lambda(B \cap (-x,x])$, $\ x \in (0,\infty)$.
I've shown that $f$ is continuous and increasing on $(0,\infty)$.
However I don't know how to formally find the limits: $\lim_{x \rightarrow \infty} f(x)$ and $\lim_{x \rightarrow 0} f(x)$. I should probably use what I've shown ?
I've come to notice that $0 \le \lambda(B \cap (-x,x]) \le 2x$ and $\lambda(B \cap (-x,x]) \le \lambda(B)$, but I don't think I should use the sandwich theorem from Calculus here ?
Also I've tried to show that for every real number $a$ in $[0, \lambda(B)]$ there exist a Borel-set $A$ such that $A \subseteq B$ and $\lambda(A)=a$.
Can someone help me out ?