Outer measure of symmetric difference

Question

Suppose $A \subset B$ and $\mu^{*}(A) = \mu^{*}(B)$.

Suppose further that $\mu_{*}(A) := 1 - \mu^{*}(E \setminus A) $ (where E is the unit with m(E)=1) is equal to $\mu^{*}(A)$

Let B be mbl.

How do I show that $\mu^{*}(B \setminus A)=0$? (Without assuming that A is mbl)

score 1 · Answer 1 · answered Nov 05 '18 at 15:59

1

It is false: the outer measure is not additive on arbitrary sets.

You can find $A\subset[0,1]$ such that $m^*(A)=m^*([0,1]\setminus A)=1$.

answered Nov 05 '18 at 15:59

Federico

Sorry, meant to add that the outer measure of A is equal to the inner measure of A (but without assuming measurability) – edwarsenal Nov 05 '18 at 16:22
Question has been edited to reflect the info I added, apologies – edwarsenal Nov 05 '18 at 16:23

1 Answers1