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 !

A function [; f: E \right arrow \mathbb{R} ;] is said to be Borel measurable if [; E ;] is a Borel set and for every [; \alpha \in \mathbb{R} ;] [; \{ x \in E : f(x) > \alpha \} ;]

I need to show that strictly increasing continuous functions map Borel sets into Borel sets.

My only idea so far is that strictly monotone functions guarantee that things are 1-1.

Let X be a random non-null measurable set of R. Can it have the property that for every other non-null measurable set A, P(A is not a subset of X) = 1?

The question involves probability space structures on the set of all non-null measurable sets of R, which are presumably very complicated objects, but I wonder if a simple direct construction might exist. The answer is no if X is restricted to be an interval, which by the way is a nice Probability Theory 101 exercise.

Probability and Stochastic Process

This rigorous course in probability covers probability space, random variables, functions of random variables, independence and conditional probabilities, moments, joint distributions, multivariate random variables, conditional expectation and variance, distributions with random parameters, posterior distributions, probability generating function, moment generating function, characteristic function, random sum, types of convergence and relation between convergence concepts, law of large numbers and central limit theorem (i.i.d. and non- i.i.d. cases), Borel-Cantelli Lemmas, well-known discrete and continuous distributions, homogeneous Poisson process (HPP), non-homogeneous Poisson process (NHPP), and compound Poisson process. This course is proof oriented. The primary purpose of this course is to lay the foundation for the second course, EN.625.722 Probability and Stochastic Process II, and other specialized courses in probability. Note that, in contrast to EN.625.728, this course is largely a non-measure theoretic approach to probability.

Compared with this course:

Theory Of Statistics

This course covers mathematical statistics and probability. Topics covered include basic set theory & probability theory utilizing proofs, transformation methods to find distribution of a function of a random variable, expected values, moment generating functions, well-known discrete and continuous distributions, exponential and location-scale family distributions, multivariate distributions, order statistics, hierarchical and mixture models, types of convergence, Delta methods, the central limit theorem, and direct and indirect methods of random sample generation. This course is a rigorous treatment of statistics that lays the foundation for EN.625.726 and other advanced courses in statistics

Problem set up:

John has two loans to repay, with initial amounts a_0 and b_0 > 0 respectively. The loans grow at a interest rate of r_t and k_t of the current amount respectively, where r_t, k_t > 0.

Assume that John has a steady flow of capital at rate C > 0. How can he most effectively pay off the loans? In the sense that he wishes to minimise, at some terminal time T, the net amount of debt he is in.

Formal problem:

Formally, let D > 0, and let us be given two continuous functions r, k: [0, T] -> [0, D].

For each pair of Borel measurable functions y, z: [0, T] -> [0, C], let A, B, K: [0, T] -> R be the (controlled) functions satisfying the differential equations

dA/dt = r(t) A - z(t)

dB/dt = k(t) B - y(t)

dK/dt = C - z(t) - y(t)

with A(0) = a_0, B(0) = b_0, K(0) = 0.

We subject the above to the following conditions:

A(t), B(t) >= 0 for all t in [0, T].

0 <= z(t) + y(t) <= C for all t in [0, T].

Thus z(t) and y(t) represent the amount of capital invested into paying off loan A and B respectively at time t, and the remainder if any, goes toward building his capital K.

Question:

Does there exist an optimal control? In the sense that the expression K(T) - A(T) - B(T) admits a maximizer among all Borel measurable y, z satisfying the above conditions.

Can we obtain a representation formula in terms of a_0, b_0, r_t and k_t?

Hello r/quant! I am new here so I apologize for any rules I may end up breaking. That said, the question I have is as follows:

1. Stochastic Calculus for Finance is something that I am enjoying but quite often I end up getting severely overwhelmed. I have a background in Electrical Engineering and I am quite comfortable in mathematics but I still find this subject extremely intimidating. What does a learner do to counter this feeling? For example, today I was learning about the Greeks and related sensitivity calculations. I am well versed with optimization but while studying this I ended up feeling so confused and overwhelmed that a one-hour long lecture took me two and a half hours. Needless to say, this is demotivating. Any advice in this matter will help!
2. Since my background stems from engineering, I have worked a lot with simulations. I, somewhat, specialize in the simulation of random processes. I see a lot of opportunities in quant finance to do this, but the notation-heavy subject makes it quite difficult. For example, I can easily simulate Markov chain behaviour (discrete-time, as of now). But when I look at the GBM of an asset as a Markov process or other Markov processes (like the Ito Process), I am unable to form a connection with conventional simulation techniques. All in all the question is:

• Is it worthwhile to pursue simulations in this field? If so, is there established literature for that?

Do your worst!

It really does, I swear!

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

I'm surprised it hasn't decade.

They’re on standbi

Buenosdillas

Pilot on me!!

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

Book Cover

https://preview.redd.it/bxiswlgn9dk71.jpg?width=433&format=pjpg&auto=webp&s=29739ad739027dd09e51e60a31a52ee3140de511

About the Author

Earned Ph.D.s in both Computer & Information Sciences and Chemistry. Senior research scientist for 10 years in pharmaceutical and medical/scientific instrument fields. Served as president and technical expert in an independent computer consulting firm for many years. Taught 20 years in universities in the US and Europe.

Prof. Dr. Donald E. Johnson

Introduction

1. This book will show that undirected naturalism lacks known scientific facts in several critical areas, and that some intelligent agent better accounts for many observations. [Donald E. Johnson: Probability's Nature and Nature's Probability (A Call to Scientific Integrity), Booksurge Publishing 2009, p4.]
2. Over time, the author began to doubt the natural explanations that had been so ingrained. It was science, and not his religion, that caused his disbelief in the explanatory powers of undirected nature in a number of key areas including the origin and fine-tuning of mass and energy, the origin of life with its complex information content, and the increase in complexity in living organisms. [Donald E. Johnson: Probability's Nature and Nature's Probability (A Call to Scientific Integrity), Booksurge Publishing 2009, p4, 5.]
3. The fantastic leaps of faith required to accept the undirected natural causes in these areas demand a scientific response to the scientific-sounding concepts that in fact have no known scientific basis. Scientific integrity needs to be restored so that ideas that have no methods to test or falsify are not considered part of science. [Donald E. Johnson: Probability's Nature and Nature's Probability (A Call to Scientific Integrity), Booksurge Publishing 2009, p5.]
4. For example, one should not be able to get away with stating "it is possible that life arose from non-life by ... " without first demonstrating that it is indeed possible (defined in the nature of probability) using known science. One could, of course, state "it may be speculated that ... ," but such a statement wouldn't have the believability that its author intends to convey by the pseudoscientific pronouncement. [Donald E. Johnson: Probability's Nature and Nature's Probability (A Call to Scienti

When I got home, they were still there.

I won't be doing that today!

You take away their little brooms

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

There hasn't been a post all year!

[Removed]

Why

f(x)= sin(1/x) if x =/= 0 and 5 if x=0

We say a function f is Borel measurable if for all [; \alpha \in \mathbb{R} ;] the set [; \{x \in \mathbb{R} : f(x) > \alpha \} ;] is Borel.

If f, g are Borel measurable, then f+g is Borel measurable.

First since f and g are Borel measurable then [; \{x \in \mathbb{R}: f(x) > \alpha \} ;] and [; \{x \in \mathbb{R}: g(x) > \alpha \} ;].

So f+g being Borel measurable, then I need to prove that [; \{x \in \mathbb{R}: f(x)+g(x) > \alpha \} ;] is Borel.

If I know that [; {x \in \mathbb{R}: f(x) > \alpha } ;]and[; {x \in \mathbb{R}: g(x) > \alpha } ;], couldn't I just say that the union of Borel measurable sets is Borel Measurable and that[; {x \in \mathbb{R}: f(x)+g(x) > \alpha} ;]` is Borel.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "