Let $B := \prod_{i=1}^n (a_i,b_i) := \{(x_1,\cdots,x_n) \in \mathbb R^n \mid \forall i: x_i \in (a_i,b_i) \}$. Can someone help me to show that $B$ is a measurable set, i.e. that if $A \subseteq \mathbb R^n$ then $m^*(A) \geq m^*(A \cap B) + m^*(A \cap B^c)$ where $ m^*(E) = \inf \left \{ \sum_{j=0}^\infty vol(B_j) : E \subseteq \bigcup_{i=0}^\infty B_i \text{ where } (B_i)_{i=0}^\infty \text{ at most countable } \right \} $ and the $B_i$ have to be open boxes. Further is $vol(B) := \prod_{i=1}^n (b_i-a_i)$ for each open box $B$.
Why open and closed boxes are measurable
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real-analysis
measure-theory
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0[For the case $n=1$](http://math.stackexchange.com/a/63044/8271) – 2012-12-29