property of lebesque outer measure

eric3353 · **Joined:** Wed, 28 May 2008 23:03:08 UTC **Posts:** 111

hey all, i am struggling with trying to work out the following property of lebesque outer measure, and was wondering if somebody could help me out:

If A and B are subsets of R, such that for some @>0, |x-y|>@ for all x in A, y in B, then m(AUB)=m(A) + m(B), where m is the lebesque outer measure.

Any assistance would be greatly appreciated,

Eric

Shadow · **Posted:** Sun, 7 Aug 2011 15:33:39 UTC

eric3353 wrote:

hey all, i am struggling with trying to work out the following property of lebesque outer measure, and was wondering if somebody could help me out:

If A and B are subsets of R, such that for some @>0, |x-y|>@ for all x in A, y in B, then m(AUB)=m(A) + m(B), where m is the lebesque outer measure.

Any assistance would be greatly appreciated,

Eric

The sets are disjoint by the given, since any $x\in A\cap B$ gives |x-y|=0<

@, (here

.)

eric3353 · **Joined:** Wed, 28 May 2008 23:03:08 UTC **Posts:** 111

the problem is, A and B are not necessarily lebesque measurable, which, if they were, the disjointness would give the desired result. but the outer measure is defined on the entire power set of R, that is, on sets that are not necessarily lebesque measurable also. the proof will have something to do with infs and covering sets, but i just cant seem to work it out yet.

outermeasure · **Posted:** Sun, 7 Aug 2011 16:10:36 UTC

eric3353 wrote:

the problem is, A and B are not necessarily lebesque measurable, which, if they were, the disjointness would give the desired result. but the outer measure is defined on the entire power set of R, that is, on sets that are not necessarily lebesque measurable also. the proof will have something to do with infs and covering sets, but i just cant seem to work it out yet.

One inequality ( $m(A\cup B)\leq m(A)+m(B)$ ) is trivial. To get the other inequality, since A,B are separated, there are disjoint opens U and V that contain A and B respectively. Take any covering $(C_i)_{i\in I}$ of $A\cup B$ (by (disjoint) Borel subsets or by (disjoint) intervals, depending on how you define the Lebesgue outer measure), and you have covering $(C_i\cap U)_{i\in I}$ of A and $(C_i\cap V)_{i\in I}$ of B, giving $m(A\cup B)+\epsilon\geq m(A)+m(B)$ for $\epsilon>0$ arbitrary.

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property of lebesque outer measure

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