Measure theory and the Intermediate Value Theorem

Question

I'm starting to study the basics of measure theory and found interesting this problem: Let us define $|E|=\int_{-\infty}^{\infty}1_E(x) \, dx$, where $E\subset \mathbb{R}$ and $1_E$ its characteristic function. Given $A\subset B \subset \mathbb{R}$, with $|A|=a<b=|B|$ and $x\in (a,b)$, is it possible to find $A\subset X \subset B$ such that |X|=x? Any ideas? Can I say this is an analogue of the Intermediate value theorem for the function $|•|$?

Answer 1

Correction due to @Brian Moehring:

Let $\varphi(x)=|A\cup(B\cap(-\infty,x))|$ for $x\in\mathbb{R}$, then $\varphi$ is continuous and $\varphi(x)\rightarrow|B|$ as $x\rightarrow\infty$ and $\varphi(x)\rightarrow|A|$ as $x\rightarrow-\infty$. Now Given $|A|<\alpha<|B|$, by IVT we have some $\varphi(c)=\alpha$ and $A\cup(B\cap(-\infty,c))\subseteq B$.