Can you produce a sequence of pairwise disjoint sets $\{E_i\}$ where

$m^{*}(\bigcup E_{i}) < \sum m^{*} (E_{i})$?

So I realize these sets cannot all be measurable, and must all have finite outer measure. Is there a specific choice of sets I should be looking for, or do I produce them from an arbitrary collection of sets? The latter seems preferable, but either method would suffice.

I know that I can pick pairwise disjoint sets that are covered by pairwise disjoint collections of intervals, and then extend each interval by, say, $\epsilon n^{-2}$. But after this, I do not know how to force a strict inequality in the final union.

  • $\begingroup$ You can pick a Vitali subset of $[0,1]$ of outer measure $1$, and consider it and its complement in $[0,1]$. $\endgroup$ – Andrés E. Caicedo May 14 '14 at 19:02
  • $\begingroup$ Of course! I was trying to do this from an arbitrary collection, but of course this is nicer. Thanks. $\endgroup$ – Johnny Apple May 14 '14 at 22:54

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