Question

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

Answer 1

Assume to the contrary that every bounded subset of E has a outer measure 0

i.e m(E) =0

Define I=[n,n+1], nz to be countable collection of disjoint bounded interval

Then we can write E as countable union of bounded subset of E

i.e E= (E I)

Now by finite sub additivity of m

0<m(E) =m(EI) m(EI)=0

which is a contradiction to our assumption

Hence there must be some nZ such that m(EI)>0