$f$ is a Herglotz-Nevanlinna function if it is analytic in the open upper half plane and the imaginary part $\Im f (z) \geq 0$ for $\Im z >0$. $f$ has the following integral representation
$$f(z) = a + bz + \int_\mathbb{R} \frac{1}{t-z} - \frac{t}{1+t^2} d \mu(t)$$
where $a \in \mathbb{R}, \ b \geq 0$ and $\mu$ is a Borel measure on $\mathbb{R}$ such that
$$\int_\mathbb{R} \frac{d \mu(t)}{1+t^2} < \infty$$
The Stieltjes inversion formula gives, that for any interval $(t_1, t_2) \subset \mathbb{R}$
$$\mu((t_1, t_2)) + \frac{\mu(\{t_1\}) + \mu(\{t_2\})}{2} = \lim_{y \to 0^+} \frac{1}{\pi} \int_{t_1}^{t_2} \Im f(t+iy) d t \tag{1}$$
My question is, why is the measure $\mu$ determined by $f$? Can the left hand side of $(1)$ be interpreted as a measure of an interval? The Wikipedia article has $\mu((t_1, t_2])$ on the left hand side. If this is the case, then I understand why the measure would be determined, but what happened to the point masses $\mu(\{t_1\})$ and $\mu(\{t_2\})$?
