An eample of a $\sigma$-finite measure on the Borels of $\mathbb{R}^2$ that isn't a Lebesgue-Stieltjes a measure?

Question

Are there any examples of a $\sigma$-finite measure on $\mathcal{B}(\mathbb{R^2})$ that is not a Lebesgue-Stieltjes measure?

By $\mathcal{B}(\mathbb{R^2})$ I mean the Borel Sets of $\mathbb{R^2}$.

Answer 1

Consider the measure that puts a unit mass at every point with rational coordinates.