I want to calculate the measure of following intervals: $$ \mu^*((a,b)), \mu^*((a,b)], \mu^*([a,b]) $$ Therefore I consider $g: \mathbb{R}\rightarrow \mathbb{R}$ and $ \mu : F^1 \rightarrow \mathbb{R}, \ \mu(A) = \sum_{i=1}^m g(b_i) -g(a_i) $ and $ A=\bigcup_{i=1}^m (a_i,b_i]$. The intervals are disjoint. g is a left-continous and monotonically increasing function. $\mu , \mu^*$ are measures.
Now I can apply Carathéodory's theorem on $\mu$ to get $\mu^*$, which is defined on $B(\mathbb{R}) := \sigma(F^1)$ $$ \mu^*(A)= \inf\left\{\sum_{i=1}^{\infty} \mu\left((a_i,b_i]\right) : A \subset \bigcup_{i=1}^m (a_i,b_i] \right\} $$
Can anybody help me calculating $$ \mu^*((a,b)), \mu^*((a,b)], \mu^*([a,b])? $$