A question about outer measure.

Question

When two sets are positively separated we know that $\mu(A \cup B)=\mu(A)+\mu(B)$. My question is what happens when their intersection is null. Will the above equation be invalid?

My Try:It has to be that the sets A or B should be non measurable or else if both are measurable then i can show that the above equation is valid. But when considering outer measures for non measurable sets i am having a lot of problems. I do not know how to assign an outer measure to non measurable sets. So if anyone can help it would be great. Thank you.

score 1 · Accepted Answer · answered Mar 15 '14 at 14:03

1

Let's use Lebesgue measure in the real line. There is a (non-measurable) subset $A$ of $[0,1]$ such that both $A$ and its complement $B=[0,1]\setminus A$ have outer measure 1. Then, of course $A \cap B = \varnothing$, but $\mu(A)+\mu(B) = 2 \ne 1 = \mu([0,1]) = \mu(A \cup B)$.

answered Mar 15 '14 at 14:03

GEdgar

111,679

can you please tell how we get a non measurable set of any size – happymath Mar 16 '14 at 08:15
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Try a Bernstein set: http://math.stackexchange.com/questions/169714/whats-application-of-bernstein-set – GEdgar Mar 16 '14 at 12:53

A question about outer measure.

1 Answers1