I am trying to show that the following two measures are identical on the Borel sigma algebra. I want to eventually apply the Caratheodory Extension Theorem. As an intermediate step, I need to show that the two measures are equal on $\mathcal{C}$, where $\mathcal{C}$ is the collection of all intervals of the form (a,b], where a,b $\in \mathbb{R}$. The two measures are:
$\mu_{1}(A) = $inf$\{\sum_{i=1}^{\infty} (b_{i}-a_{i}): A\subseteq \cup_{i=1}^{\infty}(a_{i}, b_{i}], (a_i, b_{i}] \in \mathcal{C}\}$
$\mu_{2}(A) = $lim$_{\delta \rightarrow 0}$inf$\{\sum_{i=1}^{\infty} (b_{i}-a_{i}): A\subseteq \cup_{i=1}^{\infty}(a_{i}, b_{i}], b_{i} - a_{i} \leq \delta\}$
I would a hint on how to show that $\mu_{2}(A) \leq \mu_{1}(A)$. Thanks in advance
Note that $(a_i,b_i] \in {\cal C}$ even if $b_i-a_i < \delta$.
Also, if $(a_i,b_i] \in {\cal C}$, you can add points $x_1,...,x_n$ such that $a_i<x_1<\cdots < x_n < b$ and each pair of points is less than $\delta$ apart. Furthermore, $b_i-a_i = b_i-x_n+x_n-x_{n-1}+\cdots + x_1-a_i$.
Hence for any $\delta>0$, $ \{\sum_{i=1}^{\infty} (b_{i}-a_{i}): A\subseteq \cup_{i=1}^{\infty}(a_{i}, b_{i}], (a_i, b_{i}] \in \mathcal{C}\} = \{\sum_{i=1}^{\infty} (b_{i}-a_{i}): A\subseteq \cup_{i=1}^{\infty}(a_{i}, b_{i}], b_{i} - a_{i} \leq \delta\}$.