Suppose $\mathcal{K}\subset 2^\mathbb{R}$ is such that $\sigma(\mathcal{K})=\mathcal{B}(\mathbb{R})$ and let $\mu$ and $\nu$ be measures which agree on $\mathcal{K}$, i.e. $\mu(A)=\nu(A)$ for all $A\in\mathcal{K}.$ Do these measures necessarily have to be the same?
I am teaching myself in measure-theoretic probability and for now I can only prove that the answer is yes, assuming $\mathcal{K}$ is a $\pi$-system. However, I do not know what to think when $\mathcal{K}$ is arbitrarily chosen. I would appreciate any hints.