Intersection of Two Sets Question

Jun 2014
650
54
USA
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 :(
 
Jun 2014
650
54
USA
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?
 

SDK

Sep 2016
801
544
USA
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.
 
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Aug 2012
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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.
 
Jun 2014
650
54
USA
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.