Proving Boundedness of a Subset of E with Positive Outer Measure

[sbashrawi]

Homework Statement

Show that if a set E has positive outer measure, then there is a
bounded subset of E that also has positive outer measure

Homework Equations

We can take an intersection of E with some of its covering sets. This will
give us a subset of E, but how can we be sure that it is bounded?

The Attempt at a Solution

 

[lippyka]

Let E have positive outer measure. Then there is some collection {$U_i$} of open sets such that $E \subseteq \bigcup_i U_i$ and $\sum_i m(U_i) > 0$. We can take a subset B of E by taking the intersection of E with some of its covering sets. That is, B={$E \cap U_i$}. For each element b of B, we can find a closed interval [a,b] containing b. This will give us our bounded subset B'={[a,b]: b $\in$ B}. To show that this subset B' has positive outer measure, we need to show that there exists some collection {$V_j$} of open sets with $B' \subseteq \bigcup_j V_j$ such that $\sum_j m(V_j) > 0$. We can construct this collection {$V_j$} of open sets as follows: for each b $\in$ B, pick an open interval (a,b) containing b. This gives us a collection {$V_j$} of open intervals with $B' \subseteq \bigcup_j V_j$. Since each of these open intervals have positive measure, $\sum_j m(V_j) > 0$. Thus, we have shown that B' has positive outer measure.