Proof that equality of measures on the $\pi-$system implies equality on $\sigma$ - algebra generated by it

Question

Let $\nu$ and $\nu'$ be $\sigma$-finite measures on {$E,\varepsilon$}. $K\subset \varepsilon$ is $\pi-$system, such that $\exists\, E_i\in K, E=\cup_{i=1}^{\infty}E_i $ $\nu(E_i)<\infty$. Then if $\nu=\nu'$ on $K$ $\Rightarrow$ $\nu=\nu'$ on $\sigma(K)=\varepsilon. $ $$$$I've read a lot of information, but I can't understand how to prove this statement.