Mathematics Asked by thedumbkid on February 26, 2021

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

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

Answered by Kavi Rama Murthy on February 26, 2021

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