Yep.lethe said:...you are saying that sets of isolated points are sets of measure zero.
To be fair to marcus, there actually was a good reason why he thought this and it was my post that created the confusion. I cleared things up thusly.lethe said:you did not in that post say that the notion of ioslated points and measure zero are equivalent, and i don't know why marcus would think that you did.
I suppose you could say if he'd read it more carefully, he might have said it sounds like I'm contradicting myself, which would've been true, but really, it was just an accident, as I've indicated, and I do understand measure theory.jeff said:Note that in my post,
"sets of measure zero - which can be finite or countably infinite - by definition contain only isolated points. It's only connected sets of points that can have nonzero measure: isolated points, and thus sets of isolated points since measures are additive, have measure zero simply because they have no measurable extension"
I carefully specified that "It's only connected sets of points that can have nonzero measure" as opposed to something like "connected sets of points can only have nonzero measure". Thus it should be reasonably clear that where I uncarefully used the phrase "can be", I probably intended "when", and this is in fact the case.
Yep.I agree with you that although sets of measure zero are not the same thing as sets of isolated points, the latter is example of the former. all sets of isolated points have Lebesgue measure zero.[/B]
Remarkably, this claim of mine is actually true! The classic example is the "cantor" set which is both uncountable (Edit: as opposed to countable, which isolated points are) and of measure zero. This is one of a number of bizarre results that almost made me drop physics and go into mathematics.lethe said:However, in a later post, you make this statement:
This statement is clearly mistaken. the definition of measure zero is not the same as the definition of isolated points. a set of measure zero need not contain only isolated points.[/B]
Right, the measure of lower dimensional subspaces is zero when that measure is the one defined with respect to the enveloping space of higher dimension. Okay, I understand now why you posted this. I indicated my agreement with it, but pointed out that this wasn't what marcus and I were discussing.lethe said:indeed, it was at this point that i posted a counterexample to your statement: the xy-plane in R3. Here is a set which contains many points which are not isolated, and yet is measure zero.[/B]
I'm just sorry for my part - which is nearly all - in causing the confusion.lethe said:perhaps you misspoke with your statement, and did not mean to imply that the property of "containing only isolated points" is the definition of measure zero sets, but rather just provided an example of a certain class of measure zero sets. but this is not what your wording indicated.[/B]
I think that's what actually happened.it is therefore not surprising that Rovelli, coming to the thread later, and perhaps influenced by marcus, might assume that you are confused about the difference between these two types of sets.[/B]
You could very easily be right and I wouldn't bet against it. The thing is that the curt tone is at odds with the basically conciliatory language, and this is not an uncommon way for people to acquiesce when after a long battle they realize they can no longer deny that the other guy had a point after all, but don't want to take the discussion to the "penalty phase". But again, you could very well be right. Anyway, I found this whole experience to be extremely embarrasing.i think a more reasonable assumption would be that Rovelli simply wasn't interested in continuing the dialogue.