# measure of intervals

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 $$\mu^*((a,b)), \mu^*((a,b)], \mu^*([a,b])?$$

• can you explain the notations used in your question? What tare $\mu$ and $\mu*$? WHy are you considering $g$ and what is $f$? – Mike V.D.C. May 6 at 18:42
• Sorry I will edit that – Steven33 May 6 at 18:47
• If you are attempting to typeset $\mu^*$, use \mu^*. – angryavian May 6 at 18:50