Apparent contradiction on Lebesgue-Stietjes measure?

Question

I was reviewing Folland's Real Analysis.

Given the Algebra of unions of disjoint half-open intervals (a,b].

And given an increasing, right continuous $F$.

$\mu\left(\bigcup_1^n(a_j,b_j] \right) = \sum_1^n\left(F(b_j)-F(a_j)\right)$

is a premeasure on that Algebra.

The Lebesgue-Stieltjes measure comes from this prealgebra through Caratheodory's theorem.

Now, onto my question. Let

$F(x) = \begin{equation} \begin{cases} x+1, x\geq 0\\ x, x<0 \end{cases} \end{equation} $

(sorry for the alignment)

This function is increasing and right-continuous.

For $\epsilon > 0$, we have

$\mu((0,1]) = F(1) - F(0) = 2 - 1 = 1$

$\mu((-\epsilon,1]) = F(1) - F(-\epsilon) = 2 + \epsilon = 2 + \epsilon$

But, by continuity from above, if we take $\epsilon_n$ decreasing and $\epsilon_n \rightarrow 0$

$2 = \lim_{n \rightarrow \infty} \mu((-\epsilon_n,1]) = \mu((0,1]) = 1$

How is this possible? What am i doing wrong?

Answer 1

$(-\epsilon_n ,1]$ decreases to $[0,1]$ (not $(0,1]$) and $\mu ([0,1])=\mu ((0,1])+\mu \{0\}=1+1=2$.