# Homework Help: Basic measure theory

1. Feb 21, 2010

### snipez90

1. The problem statement, all variables and given/known data
Show that if $E \subset B$ and $B \in L(\mathbb{R})$ (where L(R) denotes the family of Lebesgue measurable sets on the reals) with $m(B) < \inf$, then $E \in L(\mathbb{R})$ if and only if $m(B) = m^{*}(E) + m^{*}(B - E)$, where $m^*$ denotes the Lebesgue outer measure.

2. Relevant equations
Basic set theoretic manipulations.

3. The attempt at a solution
The forward direction follows by the definition of the outer measure. As for the reverse direction, we need to show that for any set A contained in the reals,
$$m(A) \geq m^{*}(A \cap E) + m^{*}(A - E)$$
(the less than or equal direction follows from subaddivity).
I'm not really stuck at this point, but I'm wondering about my approach from here. I simply considered the three cases where A contains B, A is contained in B and contains E, and A is contained in E. I'm fairly certain this works out in each case as it should. This seems like a pretty straightforward approach, but is there a better way? I understand this isn't exactly the most interesting of problems but it helps me to know what other tools are available to me at this point. Thanks in advance.