# Show that Lebesgue outer measure is not additive even for the finite family of pairwise disjoint sets

Let $E=[0,1]$. Show that Lebesgue outer measure on $E$ is not additive even for the finite family of pairwise disjoint sets.

I can't think about any example where only strict subadditivity holds. Is this even possible?

• Hint: your example will have to make use of non-measurable sets – Alex G. Jan 6 '16 at 14:17
• For anyone who wants to see how extreme things like this can get, see the math overflow question Is there a maximum to the amount of disjoint non-measurable subsets of the unit interval with full outer measure?. Needless to say, you (luka5z) don't need something this extreme for what you're after. – Dave L. Renfro Jan 6 '16 at 14:56
• @DaveL.Renfro Thanks for the link, I wondered about this too. It seems interesting to me. I think in my case there will be some simple example because my book is only the basic introduction. – luka5z Jan 6 '16 at 15:05