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Dec
31
answered Deduce $\lvert ab\rvert = \lvert ba\rvert$ for all $a,b\in G$ where $G$ is a group.
Dec
29
comment Are there rigorous formulation and proof of the pigeonhole principle?
To be clear, just accepting the pigeonhole principle as an obvious fact is rigorous, because it's so patently obvious. You want a formal proof, but formality and rigor are distinct notions.
Dec
26
comment Abstract alphabets by Kolmogorov??
An infinite alphabet is what it sounds like - an infinite set of symbols.
Dec
26
comment What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
@user4894 $n$-dimenional space is clearly useful in any situation in which we have $n$ real number variables, but what I object to is the general apparent assumption in mathematics that notions that are useful when $n=2,3$ and the variables are coordinates in physical space will automatically be useful in the general case.
Dec
26
comment What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
@user4894 Yes. The formula is relevant in $2$ and $3$ dimensions only because it provides the answer to problems like "how much water will I need to fill this box" or "how much canvas will I need to cover this frame" (I'm not saying mathematics has to be practical, I'm just saying that area and volume are inherently physical notions). I can't fathom why you would want to generalize it to $n$ dimensions, or even why anyone is so sure there should be a useful generalization of the idea to $n$ dimensions.
Dec
26
comment What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
@Nameless If you have time, could you be more specific in an answer?
Dec
26
comment What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
Yes, agreed.${}{}$
Dec
26
revised What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
edited title
Dec
26
comment What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
But surely at some point in defining an $n$-variable Riemann integral, we're going to use the "assumption" that the "product of the sides" formula is a good way of assigning a size to a cartesian product of intervals, no? So the very thing we're trying to justify seems to be hidden in the details.
Dec
26
comment What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
@PedroTamaroff Well, maybe what I should be asking for are applications.
Dec
26
comment What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
@PedroTamaroff But my point is that the usual definition does not appear to be meaningful. The fact that it happens to generalize a formula that means something in a very specific context doesn't make it meaningful.
Dec
26
comment What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
@Nameless If all you wanted was some definition that would allow you to integrate weird functions, I've got you covered. Define $\int f=0$ for all $f$. We don't just pick definitions because they work, we pick them because they mean something.
Dec
26
asked What is the motivation/applications for the definition of Lebesgue measure on $\mathbb R^n$?
Dec
25
comment What do bitwise operators look like in 3d?
Is that graph an actual computer rendering of XOR (as in, the computer was programmed to render the graph of XOR), or did you find out somewhere else that the graph is the Sierpinski tetrahedron and you simply rendered that? If the latter, do you have a reference?
Dec
24
comment Closed form for sine graphic rotated by 45 degrees?
@Anixx You're correct, my software was in radians rather than degrees, so that's $45$ radians or about $60$ degrees. The $45$ degree one does indeed look plausibly function-like.
Dec
24
comment Why is the naive recursive approach to defining Lebesgue measure not satisfactory?
I'm accepting this answer because I think the problem of showing consistency is the biggest gap in my approach.
Dec
24
accepted Why is the naive recursive approach to defining Lebesgue measure not satisfactory?
Dec
24
comment Why is the naive recursive approach to defining Lebesgue measure not satisfactory?
Is this squeeze property equivalent to the property "If $\mu(N)=0$ and $X\subseteq N$, then $\mu(X)=0$ ?
Dec
24
comment Closed form for sine graphic rotated by 45 degrees?
@Anixx Doesn't look like a function to me...
Dec
23
comment Why is the naive recursive approach to defining Lebesgue measure not satisfactory?
Well, the Cantor set is a complement of a disjoint union of countably many finite unions of intervals. I realize that each of those finite unions involves more and more intervals, but I don't think that's really a problem under the definition.