Measure zero (a possible problem in the reasoning)

  • #26
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The formal definition of ##\lim_{k\to \infty}g(k)\ne L## is simply ##\exists \epsilon >0 \forall N \exists k>N: |g(k)-L|\ge \epsilon##; how would you proceed from here.

OK, I see my error, thanks for pointing it out to me all of you.

Sorry to be stubborn, just when you get older you get to be stubborner.

But I agree if ##c \ne 0## then we get ##\pm \infty = 0## which is obviously a contradiction, thus ##c=0##.
So we get: ##\lim c(k+1)= \lim 0(k+1)=0##.

So foolish of me... :-(
Is there an option to delete this thread? I don't see an option somewhere.
 
  • #27
Stephen Tashi
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The formal definition of ##\lim_{k\to \infty}g(k)\ne L## is simply ##\exists \epsilon >0 \forall N \exists k>N: |g(k)-L|\ge \epsilon##; how would you proceed from here.
An outline of a proof-by-contradiction;
Assume ##\sum_{k=2}^\infty \mu(\emptyset) = 0 ## and ##\mu(\emptyset) \ne 0##

Case 1: ## \mu(\emptyset) = c > 0##
Show ##\sum_{k=2}^\infty c \ne 0##
Let ##\epsilon = 1##. For a given ##N## let ##k## be the larger of ##N+1## and the greatest integer in ##100/c##. Show ## |\sum_{i=2}^k c - 0| > \epsilon##.

Case 2: ##\mu(\emptyset) = c < 0##
Show ##\sum_{k=2}^\infty \ne 0##.
Let ##\epsilon = 1## For a given ##N## let ##k## be the larger of ##N+1## and the greatest integer in ##100/|c|##. Show ## |\sum_{i=2}^k c - 0| > \epsilon##.
 
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  • #28
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I have a further question about the definition of the symbol ##\infty## in measure theory, if we define it to satisfy: ##\infty \cdot 0 = 0\cdot \infty = 0 ## doesn't it mean that: ##0/0=\infty## for this symbol?

In which key results in measure theory do they use the multiplication condition?
 
  • #29
Math_QED
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I have a further question about the definition of the symbol ##\infty## in measure theory, if we define it to satisfy: ##\infty \cdot 0 = 0\cdot \infty = 0 ## doesn't it mean that: ##0/0=\infty## for this symbol?

In which key results in measure theory do they use the multiplication condition?
Compare with ##1.0=0=0.1\implies 1=0/0##? Even in the real case you wouldn't make such a claim.
 
  • #30
Stephen Tashi
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I have a further question about the definition of the symbol ##\infty## in measure theory,
The symbol ##\infty## is not defined in the standard approach to measure theory. It appears as an element in patterns of symbols such as "##\cup_{i=1}^\infty s_i##", or "##lim_{x \rightarrow \infty} g(x) = \infty##". Such patterns of symbols are defined, but "##\infty##" is not defined in isolation.

Informal jingles such as "##\infty \cdot \infty = \infty##" are used to remember theorems such as: If ##lim_{x\rightarrow a} f(x) = \infty## and ##lim_{x \rightarrow a} g(x) = \infty## then ##lim_{x \rightarrow a} (f(x) g(x)) = \infty##. Such theorems do not assert that ##\infty## by itself is a object that obeys certain arithmetic.

if we define it to satisfy: ##\infty \cdot 0 = 0\cdot \infty = 0 ##
There is no such definition in measure theory. The formal statement about limits implied by the jingle "## 0 \cdot \infty = 0##" is not, in general, true.
 
  • #31
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Compare with ##1.0=0=0.1\implies 1=0/0##? Even in the real case you wouldn't make such a claim.
But ##\infty## isn't a real number.

In the real number case you would divide by ##1## for example. So you say that this symbol satisfies also: ##0 = 0/\infty##?
 
  • #32
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The symbol ##\infty## is not defined in the standard approach to measure theory. It appears as an element in patterns of symbols such as "##\cup_{i=1}^\infty s_i##", or "##lim_{x \rightarrow \infty} g(x) = \infty##". Such patterns of symbols are defined, but "##\infty##" is not defined in isolation.

Informal jingles such as "##\infty \cdot \infty = \infty##" are used to remember theorems such as: If ##lim_{x\rightarrow a} f(x) = \infty## and ##lim_{x \rightarrow a} g(x) = \infty## then ##lim_{x \rightarrow a} (f(x) g(x)) = \infty##. Such theorems do not assert that ##\infty## by itself is a object that obeys certain arithmetic.



There is no such definition in measure theory. The formal statement about limits implied by the jingle "## 0 \cdot \infty = 0##" is not, in general, true.
Well, @Math_QED said in post number 8 (I see the irony :-)) in this thread that in Measure theory that it's indeed satisfy this relation.

BTW, for those who don't see the irony I remind you of the following joke:
:cool:
 
  • #33
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There is no such definition in measure theory.
I beg to disagree. Here are standard references for measure theory that all mention it.

(1) Donald Cohn's "Measure theory" -> Appendix A, p380
(2) Follands "Real analysis" -> p11
(3) Rudin's "Real and complex analysis" -> p18

Indeed, such a definition is useful in situations like this:

Consider the product ##\mu \otimes \nu## of two (##\sigma##-finite measures) ##\mu, \nu##. Such measures are completely determined by the property ##\mu\otimes \nu(A \times B) = \mu(A) \nu(B)##.

Now, consider the case where ##A = \emptyset, \nu(B) = +\infty##. Then the above condition reads ##\mu\otimes \nu(\emptyset) = \mu\otimes \nu(\emptyset \times B) = \mu \otimes \nu (A \times B) = \mu(\emptyset) \nu(B) = 0 . \infty## and the definition ##0. \infty = 0## is needed naturally, unless you always want to specify the exceptions.

You literally never run into trouble, as long as you don't make up rules yourself.

Really confused as why someone would say that things like ##\infty + \infty## are not defined in measure theory. Not only are they rigorously defined, they are also necessary!

For example, consider the counting measure ##\sharp## on ##\mathbb{R}##. Then

##\sharp(\mathbb{Q} \cup \mathbb{Q}^c) = \sharp(\mathbb{Q})+ \sharp(\mathbb{Q}^c)= \infty + \infty##

Oh no! It's ##\infty + \infty##! We are doomed! Unless we, maybe, define ##\infty + \infty := \infty##? It is pure nonsense to say that extended reals are not rigorously defined. One just has to be careful with it. Context is everything.
 
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  • #34
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The symbol ##\infty## is not defined in the standard approach to measure theory. It appears as an element in patterns of symbols such as "##\cup_{i=1}^\infty s_i##", or "##lim_{x \rightarrow \infty} g(x) = \infty##". Such patterns of symbols are defined, but "##\infty##" is not defined in isolation.

Informal jingles such as "##\infty \cdot \infty = \infty##" are used to remember theorems such as: If ##lim_{x\rightarrow a} f(x) = \infty## and ##lim_{x \rightarrow a} g(x) = \infty## then ##lim_{x \rightarrow a} (f(x) g(x)) = \infty##. Such theorems do not assert that ##\infty## by itself is a object that obeys certain arithmetic.



There is no such definition in measure theory. The formal statement about limits implied by the jingle "## 0 \cdot \infty = 0##" is not, in general, true.
I remember Royden working under that assumption.
 
  • #35
Stephen Tashi
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I beg to disagree. Here are standard references for measure theory that all mention it.
We have to distinguish between defining something versus mentioning it . -

Using symbol as part of an aggregation of other symbols is mentioning it. This doesn't imply that a definition for the entire aggregation also gives a definition for the individul symbols that compose it.

I agree that the symbol "##\infty##" appears in writings on measure theory. I do not agree that this symbol has one specific definition that applies in all the places it appears.

Likewise, symbols such as "##\rightarrow##" appear in writings on measure theory. In that sense they are mentioned, but not given a definition in their own right.


(3) Rudin's "Real and complex analysis" -> p18

which is a book, I happen to have. Rudin says on p18

For any ##E \subset X##, where ##X## is any set define ##\mu(E) = \infty ## if ##E## is an infinite set and let ##\mu(E) ## be the number of of points in ##E## if ##E## is finite.
This is not a definition of "##\infty##".

It could be considered to define the collection of symbols "##\mu(E) = \infty##" to mean "##E## is an infinite set".

Prior to that, on page 8, Rudin uses the symbol "##\infty##" in the aggregation of symbols "##f:X \rightarrow [-\infty, \infty]##" , so if he's writing mathematics properly, he wouldn't used a concept on page 8 and then wait till page 18 to define it.

I think Rudin takes for granted that the reader will interpret ##\infty## in the context of the "extended real number line" in some places in his text. However, the arithmetic of an abstract number line does not explain why aggregations of symbols such as "##lim_{x \rightarrow a} f(x) = L##" and "##lim_{x \rightarrow a} f(x) = \infty##" require different definitions, one with the condition ##|f(x) - L| < \epsilon## and the other with the condition ## f(x) > r ##.

Indeed, such a definition is useful in situations like this:
However, I see no definition of "##\infty##" that is the same definition in all situations.

One just has to be careful with it. Context is everything.
I agree that the meaning of the symbol "##\infty##" is dependent on the context in which it used. My point is that it has no universal definition.
 
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  • #36
Math_QED
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We have to distinguish between defining something versus mentioning it . -

Using symbol as part of an aggregation of other symbols is mentioning it. This doesn't imply that a definition for the entire aggregation also gives a definition for the individul symbols that compose it.

I agree that the symbol "##\infty##" appears in writings on measure theory. I do not agree that this symbol has one specific definition that applies in all the places it appears.

Likewise, symbols such as "##\rightarrow##" appear in writings on measure theory. In that sense they are mentioned, but not given a definition in their own right.





which is a book, I happen to have. Rudin says on p18



This is not a definition of "##\infty##".

It could be considered to define the collection of symbols "##\mu(E) = \infty##" to mean "##E## is an infinite set".

Prior to that, on page 8, Rudin uses the symbol "##\infty##" in the aggregation of symbols "##f:X \rightarrow [-\infty, \infty]##" , so if he's writing mathematics properly, he wouldn't used a concept on page 8 and then wait till page 18 to define it.

I think Rudin takes for granted that the reader will interpret ##\infty## in the context of the "extended real number line" in some places in his text. However, the arithmetic of an abstract number line does not explain why aggregations of symbols such as "##lim_{x \rightarrow a} f(x) = L##" and "##lim_{x \rightarrow a} f(x) = \infty##" require different definitions, one with the condition ##|f(x) - L| < \epsilon## and the other with the condition ## f(x) > r ##.



However, I see no definition of "##\infty##" that is the same definition in all situations.



I agree that the meaning of the symbol "##\infty##" is dependent on the context in which it used. My point is that it has no universal definition.
I was referring to defining of ##0. \infty = 0 = \infty.0##, not the symbol ##\infty## itself.

For me, the extended real number ##\infty## is just an element disjoint from ##\mathbb{R}## satisfying ##\infty \geq x## for all ##x \in \mathbb{R}##.

Also, the definitions of the limit can all be unified if one considers the order topology on ##[-\infty,\infty]## so while the definitions look different at first, they are a subset of a broader definition. See Rudin's "Principle of mathematical analysis", definition 4.33, p98.

I agree with your statement that there is no universal usage of the symbol infinity, but its position in measure theory is well-established, which was my point.
 
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  • #37
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About 4 decades ago, my mathematics professor, when I tried to object to the inconsistency of seeing the slices of infinitesimal width as non-zero when summing them and as zero when they were considered individually, introduced me to the concept of 'treated as zero' -- the important thing was to get a usable area under the curve -- you need to know how much paint to purchase . . .
 
  • #38
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About 4 decades ago, my mathematics professor, when I tried to object to the inconsistency of seeing the slices of infinitesimal width as non-zero when summing them and as zero when they were considered individually, introduced me to the concept of 'treated as zero' -- the important thing was to get a usable area under the curve -- you need to know how much paint to purchase . . .
There is a rigorous account of this is non-standard analysis. No issue with ant ghosts of departed quantities ;).
 
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  • #39
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ant ghosts of departed quantities
That seems to me to be a nice way to term the infinitesimals that we routinely discard.
 
  • #40
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In my view, saying that the infinite set of integers has measure zero is an abuse of language -- I think that anything that is not non-existent is more than zero . . .
 
  • #41
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In my view, saying that the infinite set of integers has measure zero is an abuse of language -- I think that anything that is not non-existent is more than zero . . .
It just means that to the effects of the value of an integral, it makes no difference. But if you think about it, you can ask similar questions in every single branch of knowledge.
 
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  • #42
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It just means that to the effects of the value of an integral, it makes no difference. But if you think about it, you can ask similar questions in every single branch of knowledge.
In this instance, I didn't ask a question; I merely objected to the linguistic abuse of the term 'zero' -- in my view, we should say 'negligible' instead of 'zero', in order to reserve 'zero' for the non-existent and the impossible.

If you admit that I can choose a number at random from the unit interval, the probability that the chosen number is .5 is negligible but't it's not really zero; the probability that the number is 2 is really zero because 2 is absolutely outside the unit interval and therefore impossible to choose within it.

Please go ahead and scold people for not having learned real number analysis, but also please don't rob people of the legitimate meanings of words, including that zero means nil.
 
  • #43
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In this instance, I didn't ask a question; I merely objected to the linguistic abuse of the term 'zero' -- in my view, we should say 'negligible' instead of 'zero', in order to reserve 'zero' for the non-existent and the impossible.

If you admit that I can choose a number at random from the unit interval, the probability that the chosen number is .5 is negligible but't it's not really zero; the probability that the number is 2 is really zero because 2 is absolutely outside the unit interval and therefore impossible to choose within it.

Please go ahead and scold people for not having learned real number analysis, but also please don't rob people of the legitimate meanings of words, including that zero means nil.
I you reject that the probability of choosing a specific real value from an interval is zero, then you must propose another real value for the probability. Probability is defined to be a real number from 0 to 1 inclusive. So if you reject zero for the probability than you must choose another number. Any other choice you make will simply be wrong.
 
  • #44
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I[f] you reject that the probability of choosing a specific real value from an interval is zero, then you must propose another real value for the probability. Probability is defined to be a real number from 0 to 1 inclusive. So if you reject zero for the probability th[e]an you must choose another number. Any other choice you make will simply be wrong.
I choose 'infinitesimally more than zero' instead of zero for the probability -- in my view, only the impossible has zero probability -- the probability that some number within the interval will be the number is 1, wherefore every number within that interval has a positive non-zero chance; only numbers outside of the interval genuinely have zero chance.
 
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  • #45
Stephen Tashi
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Since discussions of personal theories are not allowed on the forum, we should return to the main topic of the thread - if there is anything further to say about it.
 
  • #46
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Since discussions of personal theories are not allowed on the forum, we should return to the main topic of the thread - if there is anything further to say about it.
I think that you in post #35, and @Math_QED in #36, (and others in other posts) said pretty much enough to fully respond regarding the main topic of this thread, but the topic itself invites the possibilities of dissent or acquiescence or somewhat nuanced difference of opinion, which I think are not necessarily things that we should regard as prohibited here.
 
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  • #47
DrClaude
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Time to close the thread. Thank you to all that have participated.
 
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