Borel regular measure Puns

Hi, I'm attempting solve this problem. I believe I summarized it well here: https://math.stackexchange.com/questions/3566574/prove-that-a-lebesgue-measurable-function-is-almost-everywhere-equal-to-a-borel

Help would be greatly appreciated, thanks!

So I am working my way through Axler's Measure, Integration & Real Analysis free textbook (http://measure.axler.net/MIRA.pdf) but I find myself confused by the information presented on p18 which states that we cannot show the outer measure of a closed interval [a,b] is b-a without the Heine-Borel theorem.

In the book he shows an upper bound can be easily found since [a, b] subset (a - epsilon , b + epsilon) for any epsilon>0 and therefore abs([a, b]) <= b-a. However he then states that a lower bound cannot be found as easily, and gives some confusing justification which involves supposing the reals were countable?

But since we already know of the monotonicity property of the outer measure and given that (a, b) subset [a, b] would this not provide us with the lower bound? I am not sure what I am missing that makes this impossible.

Any advice would help, thanks!

I am trying to figure out if (-inf, x] is in a Sigma Algebra, and I know Borel sets are always in a sigma algebra. My intuition is yes, but I am just seeking confirmation. Thank you!

I'm finding the exercises for chapter 2 pretty challenging. Exercise 12 asks you to show that every compact set is the support of some Borel measure, that is, an open set has nonzero measure if and only if it has a nonempty intersection with the set.

For closed intervals, you can cook up something using Lebesgue measure, and for singleton sets {z} you have The Dirac ditribution, where an open set has measure 1 if it contains the point, and measure 0 if not.

But there are more complicated compact sets, such as the Cantor set, which has no intervals or isolated points (it is totally disconnected, but every point is a limit point of the set). The Cantor set has the Cantor distribution, which is generated by the Cantor function c(x) (you should look this one up, it is monotonically increasing from zero to one, is a.e. constant, but is continuous, a really strange function), so that the measure of an open interval (a,b) is c(b) minus c(a).

I first thought I could come up with such a function in the general case, using the construction of the Cantor function as the uniform limit of a sequence of piecewise-linear monotonically increasing functions, when it occurred to me that you couldn't get the Dirac distribution using such a function, since the measure "jumps" at the point z. But if you use the upper semicontinuous characteristic function chi(x) on [z,1], then define tthe measure of (a,b) as the lower limit of chi at b minus chi(a), then that works.

Of course, you have to show that you can construct such a function in the general case and that it works. This just gets messier and messier.

Hi,

I'm going through a book that is showing me Falconer's Measurable Chromatic Number of the Plane result. At the same time I'm learning the measure theory required. I understand Lebesgue measure and measurable sets making a sigma algebra, but the definitions for Borel sets seem inconsistent. I think the book is defining them in a non-standard way so it's adding to my confusion. Here it is:

It is shown in all measure theory textbooks that the collection of all measurable

sets is a σ-field. The intersection of all σ-fields containing the closed sets is a σ-field

containing the closed sets, the minimal such σ-field with respect to inclusion. Its

elements are called Borel sets. Since closed sets are measurable and the collection

of all measurable sets is a σ-field, it follows that all Borel sets are measurable.

I understand complementation and countable union, but I don't understand how they are used in relation to borel sets. Is it just a way setting certain parameters for sets?

And then the Lebesgue measure sounds to me just like the definition of length with countable additivity and non-negativity added. My professor made it sound like the Lebesgue measure only applies to borel sets or something. Is this the case? if not, what is the relation between Borel sets and the Lebesgue measure?

thank you !

I imagine it's derived indirectly by comparing the physical and/or chemical properties of unprocessed and processed samples, but what about other microscopic structures, like intestinal lining, that don't have as straightforward properties to test?

And more to the point, what do we mean when we say "surface"? Where's the cutoff for various standards (chemical, biological, or architectural) where surface ends and interior begins?

This is the best tl;dr I could make, original reduced by 63%. (I'm a bot)

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> PURPOSE: The aim of this study was to explore fatigue, sleep, physical activity, fitness markers, and fitness outcomes in cannabis and cannabidiol users.

> Participants were cannabis users, cannabidiol users, and control.

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> These findings provide support for future randomized, controlled clinical trials examining the effects of cannabis and cannabidiol or other health and fitness outcomes.

Summary Source | FAQ | Feedback | Top keywords: Cannabis^#1 group^#2 Cannabidiol^#3 fatigue^#4 sleep^#5

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Alguém me explica isso.

Você olha pra casos como o PC Siqueira, por exemplo, o cara praticamente morreu na mídia.

Já o Nego do Borel acaba de ser expulso de A Fazenda mediante investigação por estupro, e lá no Instagram dele ainda tem gente defendendo o cara.

Firewood is somewhat regularly sold in a Klafter, nowadays usually meaning 3 m³. Historically it was a volume measurement of one cubic (length)Klafter. A length Klafter was 6 Austrian feet (316mm).

The liquid volume measurements of Pfiff (177ml) and Seidel (354ml) are exclusively used for beer nowadays, historically they used them for different luquids and for some reason grain. In Germany and western Austria I think, a Maß (1,4l) is common for beer.

I heard that tailors still sometimes use the Elle, but it has barely anything to do with the historical one (originally the length of an adult forearm, so it was very different in each city, the Viennese Elle was 77cm). I think nowadays it's just a short way to say "50cm", because the german standard Elle was about 50cm.

I think in some regions they sometimes use a Joch for areas, it was originally the size of a field that an ox could plough in a day.

Of course all of these are only used informally these days and are nowadays fixed to metric measurements.

btw, he didn't tell me what kind of tool tbh

SOLVED:

According to the comments, it is true that you can break the carbon when you adjust the saddle using regular tools like hex key.

So, I have bought the torque wrench.

#Link to Bout

#Summary

What happens when you get two of the best offensive fencers in the world in Borel and Park going head to head?

You get something that looks more like a Sabre bout than an epee bout. This one was a thriller, a 15-14 win for Park that lasted just 131 seconds, had 25 unique touches, and multiple lead changes.

• The average touch has taken 17.1 seconds to set up from what I’ve recorded thus far. This bout, the longest touch took 14 seconds to score. The average touch here took 5.24 seconds per touch— by far the fastest bout I’ve recorded to date.

• Borel did not attempt to pull Park to his side of the strip a single time in this bout. His goal was to pursue, collapse distance, and keep a constant threat on Park. It was pretty much a 50/50 success to failure ratio. Borel had the lead for 10 touches, Park had the lead for 10 touches, and they were tied for 5. Straight up “YOLO bout.”

• So many fast one tempo actions in the box! Both of these guys are absolute speed demons. 15/25 touches occurred in 5 seconds or less. In fact, 7/8 of the final touches were in 5 seconds or less.

• No surprise given the style/speed/preferences of these two fencers, but Borel attacked one tempo 5/10 of his single lights. Park was one tempo 6/11 of his single lights.

• Borel’s predominate action was a direct lunge (4x). For Park, also a lunge (6x)

• On a side note, I’ve now recorded 70 single lights for Borel. He averages 12.5 seconds per touch. He’s a fast moving train that’s hard to stop.

#Touch by Touch Breakdown

https://preview.redd.it/rufdag0fcpm71.jpg?width=284&format=pjpg&auto=webp&s=d4da9fb0f796ed7081490fe3452927bd8842acf2