Statement 1.7 from Fractal Geometry by Falconer
I am reading through Fractal Geometry Mathematical Foundations and Analysis by Falconer and I am not very familiar with measure theory.
While reading through equation 1.7, I have been stuck trying to figure out if I am just dumb or if the textbook is wrong. $A_\delta$ is the delta neighborhood of a set. I believe that if delta is decreasing, then the Borel sets are also decreasing. I think the true statement is likely $$\lim_{\delta\to 0}\mu\left(A_\delta\right) = \mu\left(\bigcap_{\delta > 0}A_\delta\right) \stackrel{?}{=} \mu(A)$$