Prove limit of volume is volume of limit for a sequence of compacts

Question

Let $K_n$ be compact sets $n\in\mathbb{N}$ with $K_n\subset K_{n+1}$ for all $n\in\mathbb{N}$ and such that $A=\cup_{n=1}^\infty K_n$ has well-defined volume (volume is the way I call Jordan measure). I need to prove that$$v(A) = \lim_n v(K_n)$$ As you can see, the exercise doesn't guarantee the $K_n$ have well-defined volume, but it has to be true for the proof to make sense (at least for all $n$ greater than a certain $k\in\mathbb{N}$). If I assume the compacts have volume, then the proof is easy using some previous results. However, I'm not using the fact that the sets are compact, so probably this information is necessary to prove they have well-defined volume. Is there a way I can prove that there exists an $N\in\mathbb{N}$ such that for all $n>N$ : $K_n$ has volume?