# Show that if $m^∗(E) < ∞$ and there exist intervals $I_1, . . . , I_n$ such that $m∗E(∆(∪_{i=1}^{n}I_i))< ∞$, then each of the interval are finite

Show that if $$m^∗(E) < ∞$$ and there exist intervals $$I_1, . . . , I_n$$ such that $$m^∗(E(∆(∪_{i=1}^{n}I_i)))< ∞$$, then each of the interval $$I_i$$ is finite.

I have been asked to prove this and I am thinking along the lines of Littlewood's First Principle, but I haven't made much progress yet. Any help would be appreciated

• How do you define the outer measure $m^*$? – Keen-ameteur 2 days ago
• @Keen-ameteur I have defined $m^*$ as the infimum of the length of the union of the open intervals which contain that set – thedumbkid 2 days ago

There exists an open set $$U$$ such that $$E \subseteq U$$ and $$m(U) . We can write $$U$$ as countable disjoint union of open intervals $$(a_i,b_i)$$. Now $$\bigcup_{i=1}^{n} (a_i,b_i)) \subseteq (E\Delta \bigcup_{i=1}^{n} (a_i,b_i)) \cup E$$ so $$m(\bigcup_{i=1}^{n} (a_i,b_i))<\infty$$. This implies that each of the $$I_i$$'s is a finite interval.