My Math Forum Intersection of Two Sets Question

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May 2nd, 2019, 07:35 PM   #11
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Quote:
 Originally Posted by Maschke Didn't you say it was 1?
I said the intersection between a Vitali set and the rationals would have a cardinality of 1. You said it had a measure of 0 and I'm fine with that too, but that's wholly irrelevant. Here I thought maybe you were just alluding to the fact that you thought the measure of the relevant intersection would be 0 since the measure of $\mathcal{C}$ is zero, but now I realize you really didn't have a clue what this thread was about. double

 May 2nd, 2019, 07:39 PM #12 Senior Member   Joined: Jun 2014 From: USA Posts: 525 Thanks: 40 Let's pose a more generalized version of the question for you then. Are there sets A and B where A $\cap$ B = A, the measure of A is undefined, and the measure of B is 0?
May 2nd, 2019, 08:33 PM   #13
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Quote:
 Originally Posted by AplanisTophet now I realize you really didn't have a clue what this thread was about.
I'm truly sorry I wasted my time on you tonight.

May 2nd, 2019, 08:38 PM   #14
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Quote:
 Originally Posted by AplanisTophet I get it now, you were neglecting the possibility that the measure of the intersection between $V$ and $\mathcal{C}$ could be undefined, especially if the intersection were to equal $V$. You can't just assume that the intersection would have a measure of $0$ simply because $\mathcal{C}$ does as this says nothing of whether or not the measure of the intersection may be undefined.
He is neglecting that possibility because it isn't a possibility. It's a fundamental fact that if $A$ has (Lebesgue) measure zero, then every subset of $A$ is measurable and also has measure 0. In fact, this is true of any complete measure which follows as an easy consequence of the Carathéodory (sp?) completion procedure.

Last edited by skipjack; May 2nd, 2019 at 08:58 PM.

May 2nd, 2019, 11:09 PM   #15
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Quote:
 Originally Posted by AplanisTophet Let's pose a more generalized version of the question for you then. Are there sets A and B where A $\cap$ B = A, the measure of A is undefined, and the measure of B is 0?
Hey man I'm sorry I overreacted. That silly goose remark got to me for some reason. You're right, you said cardinality and I misread it. All the best.

May 8th, 2019, 06:11 PM   #16
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Quote:
 Originally Posted by Maschke Hey man I'm sorry I overreacted. That silly goose remark got to me for some reason. You're right, you said cardinality and I misread it. All the best.
No worries. You told me what a Vitali set was in the first place.

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