# Lebesgue measure

1. Sep 17, 2009

### atwood

I've seen the definition that an outer Lebesgue measure is defined as
$$m_n^* (A) = \inf \left\{ \sum_{k=1}^{\infty} l(I_k) \, \left| \, A \subset \bigcup_{k=1}^{\infty} I_k \right}$$
where Ik are n-dimensional intervals and l(Ik) is the geometric length.

It is not actually clear to me if A has to be a proper subset. That is, does
$$A \subset \bigcup_{k=1}^{\infty} I_k$$
actually mean
$$A \subseteq \bigcup_{k=1}^{\infty} I_k$$
or
$$A \subsetneq \bigcup_{k=1}^{\infty} I_k$$
?

2. Sep 17, 2009

### Dick

Why do you think the distinction between those two cases is important? Show that the sum of l(I_k) can be made arbitrarily close to the sum of the lengths of intervals J_k where each I_k is a proper subset of J_k. That's a good exercise.