Measure zero (a possible problem in the reasoning)

[member 587159]

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.

 

[sysprog]

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 . . .

 

[WWGD]

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 ;).

 

[sysprog]

ant ghosts of departed quantities

That seems to me to be a nice way to term the infinitesimals that we routinely discard.

 

[sysprog]

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 . . .

 

[WWGD]

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.

 

[sysprog]

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.

 

[PAllen]

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.

 

[sysprog]

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.

 

[Stephen Tashi]

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.

 

[sysprog]

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.

 

[DrClaude]

Time to close the thread. Thank you to all that have participated.