[math-fun] Losing sleep because...
For reals R, card(R^n)=card(R) yet R is measure zero in R^n. This is something of a paradox, yet: R^n=(R^(n-1)/p)*R + p*R, with p a lone point of R^(n-1), and p measure zero in R^(n-1). This sounds okay(?). What’s more troubling is reading on stack exchange and physics forum that Lebesgue integration allows, sometimes requires (?), 0*infinity=0? “I’m on the outside, looking inside / what do I see? / Much confusion, disillusion / all around me.” Arggggg.......
Does it bother you, Brad, that if S is a Cantor set in R, card (S) = card (R) but S is measure zero in R? On Thu, Aug 8, 2019, 4:18 AM <bradklee@gmail.com> wrote:
For reals R, card(R^n)=card(R) yet R is measure zero in R^n. This is something of a paradox, yet:
R^n=(R^(n-1)/p)*R + p*R,
with p a lone point of R^(n-1), and p measure zero in R^(n-1). This sounds okay(?).
What’s more troubling is reading on stack exchange and physics forum that Lebesgue integration allows, sometimes requires (?), 0*infinity=0?
“I’m on the outside, looking inside / what do I see? / Much confusion, disillusion / all around me.”
Arggggg.......
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Or does it bother you, Allan, that Hofstadter's butterfly is a Cantor set, yet it can be measured in a physical laboratory? ( see also: https://www.youtube.com/watch?v=1JdS-1-yYu8 , also at times 1:03:00 and 1:09:00 ) Another quandry is that Rationals Q are dense in R, yet measure zero. Practically speaking, a numerical integral on the unit interval [0,1] is usually taken over a Q subset (precision-limited decimals), but we have already said that such a set "contributes nothing"! See for example: In[]:= Total[Table[N[Sin[Pi/2*RandomInteger[{0, 1000000}]/1000000 ]*Pi/2], {1000000}]]/1000000 Out[]:=1.00058 --Brad On Thu, Aug 8, 2019 at 3:30 AM Allan Wechsler <acwacw@gmail.com> wrote:
Does it bother you, Brad, that if S is a Cantor set in R, card (S) = card (R) but S is measure zero in R?
I'm not sure why Hofstadter's butterfly should bother me. Anyway, no, it doesn't. On Thu, Aug 8, 2019 at 1:07 PM Brad Klee <bradklee@gmail.com> wrote:
Or does it bother you, Allan, that Hofstadter's butterfly is a Cantor set, yet it can be measured in a physical laboratory? ( see also: https://www.youtube.com/watch?v=1JdS-1-yYu8 , also at times 1:03:00 and 1:09:00 )
Another quandry is that Rationals Q are dense in R, yet measure zero. Practically speaking, a numerical integral on the unit interval [0,1] is usually taken over a Q subset (precision-limited decimals), but we have already said that such a set "contributes nothing"!
See for example:
In[]:= Total[Table[N[Sin[Pi/2*RandomInteger[{0, 1000000}]/1000000 ]*Pi/2], {1000000}]]/1000000 Out[]:=1.00058
--Brad
On Thu, Aug 8, 2019 at 3:30 AM Allan Wechsler <acwacw@gmail.com> wrote:
Does it bother you, Brad, that if S is a Cantor set in R, card (S) = card (R) but S is measure zero in R?
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