Measure Definition and 1000 Threads

Hey i'm given a measure on $\Omega =\mathbb{N}, A \subset \mathbb{N}$ and should check additivity as well as $\sigma$-additiviy. Additionally i should give an example for a set where $\mu(A)$ is not defined: $\mu(A)=\lim_{n \rightarrow \infty} \frac{1}{n} \xi\mid _\mathbb{N} (A \cap...

Do we essentially measure its wavelength and calculate the frequency? I thought we figured out frequencies by refracting the light into its many frequencies - (ie, like we see when looking at a Doppler shift). But, refraction formulas are based on VELOCITY, not wavelength and light travels at...

Homework Statement [PLAIN]http://www.album.com.hk/d/1546698-2/Untitled-1.jpg Homework Equations Every measurable set in R can be written as a difference of Gdelta set and a set of measure zero. Every open set in R is just a countable union of disjoint intervals. The Attempt at a...

Why do we measure the angle of incidence or reflection (or say refraction even) w.r.t normal to the surface and why not w.r.t the horizontal surface?

Hello all, I have a few questions in my mind: 1) \lim_{n\rightarrow \infty}[0,n) = \cup_{n\in\mathbb{N}}[0,n) = [0,infty) holds, and for \lim_{n\rightarrow \infty}[0,n] = \cup_{n\in\mathbb{N}}[0,n] = [0,infty) is also true? It should not be [0,infty] , am I correct? 2)...

see the pic [PLAIN]http://store2.up-00.com/Oct10/opW54159.jpg

Hey guys, below is a small question from introductory measure theory. Maybe be completely wrong on this, so if you could point me in the right direction I'd really appreciate it. Claim: Let B=\mathbb{Q} \cap [0,1] and \{I_k\}_{k=1}^n be a finite open cover for B. Then \sum_{k=1}^n m^*(I_k)...

Homework Statement Let (X , M, m) be a certain measure space and {En} sets in M with the property: \underline{lim} (Ek) = \overline{lim}(Ek) = E prove that lim m(Ei) exists and = m(E) as n approaches infinity. Homework Equations The Attempt at a Solution i solved the problem...

Is there a "standard" probability measure one would use for the set of smooth real-valued functions on [a, b]? My intuition is picturing a setup where you cut out shapes in the x-y plane, and then the set of functions whose graphs are contained in that shape have a measure proportional to the...

This is a pretty elementary question but I had it on a quiz and it made me think... does a speedometer measure a scalar or vector quantity? I answered that it measures a vector quantity my rational being that it is the instantaneous speed in a forward direction, always. It doesn't matter if...

On this webpage http://www.mathpages.com/home/kmath241/kmath241.htm" are these words of Maxwell: "The only practicable method of determining directly the relative velocity of the aether with respect to the solar system is to compare the values of the velocity of light deduced from the...

Dear all, I have a piece of brass material at high temperature (around 300 Celsius) and I would like to know how can I measure the heating power stored by this piece. the piece is designed in a way that we can flow a fluid through it. For low temperatures ( below 100 Celsius) it is maybe...

Homework Statement Show that there is a continuous , strictly increasing function on the interval [0, 1] that maps a set of positive measure onto a set of measure zero Homework Equations The Attempt at a Solution I need to find a mapping to a countable set or cantor set but I...

Is the integral over a set of measure zero always equals to zero? Can the integral be undefined?

Hi, How can I measure the capacitance of a capacitor, using the following: a battery, a digital voltmeter (which records voltage as a function of time), a digital ammeter (which records current as a function of time), an industrial capacitor, two copper plates, and several...

If we have two red balls and we are to choose one it is true that the probability measure of picking a red ball is 1. In this case it is understood that the sample space only contains a red ball and this is because events in the sample space are to be disjoint. But how come this is not the...

If I squeezed someone with a hug, how would I measure the force

Hi all I am struggling to see the difference between Convergence in probability and Almost surely convergence of a sequence of random variables. From what i can see Almost surely convergence of Sequence of Random variables is very similar to pointwise convegence from Real analysis. I am...

Homework Statement Let A and B be bounded sets for which there is \alpha > 0 such that |a -b| \geq\alpha for all a in A and b in B. Prove that outer measure of ( A \bigcup B ) = outer measure of (A) + outer measure of (B) Homework Equations We know that outer measure of the union is...

i have no idea of how to measure a berry phase take a 1/2 spin in an external magnetic field it is well known that this system can exhibit berry phase if the external magnetic field B takes a loop on the sphere. but how to measure the berry phase? how to separate the berry phase from...

Homework Statement In a triangle, two sides that measure 6 centimeters and 10 centimeters form an angle that measures 80 degrees. Find to the nearest degree, the measure of the smallest angle in the triangle. Homework Equations A2 = B2 + C2 – 2bc Cos A The Attempt at a Solution...

In general, on earth, angle is easier to measure with a reference point, but what about measuring something in space (with no reference point?)

Hi all, I was thinking about measuring the temperature of an arc discharge. The size of the discharge is very small, in the order of micrometer. So it is not possible to put a thermocouple near the it. Thank you very much.

hello people can anyone help me...? here is my objective... To measure how long a person stays on the air, when it jumps, until it reaches the ground at the same position. Given: Weight of the person Force exerted Find: Time Height of the jump ...plsss

Homework Statement Find the integral of the function f(x)=3 when x is rational and 2 when x is irrational on the interval [0,1]. Homework Equations The Attempt at a Solution So I partition [0,1] into two disjoint sets A and B. A=[0,1] \cap Q and B=[0,1] \cap Q^{c}. Now the...

Considering they will eventually be colliding gold or lead ions, will they measure or be able to measure the gravitational impact of the collisions? Just curious since the velocity will be so extreme. I would think Tevatron would already have something to say about this, but I'm unsure of...

Hi All, I am a new phd student in engineering, working in signals analysis in neuroscience who seems to be doing a lot of work in statistics and probability theory. My uni is offering a course in measure theory. The course profile says: "The course is an introduction to measure theory and...

Homework Statement Given a measure space (\mathbb{R}, \mathcal{B}(\mathbb{R}),\mu) define a function F_\mu : \mathbb{R} \to \mathbb{R} by F_\mu(x) := \mu( (-\infty,x] ). Prove that F_\mu is non-decreasing, right-continuous and satisfies \displaystyle \lim_{x \to -\infty} F_\mu(x) = 0. (Right...

I just finished a course where we discussed concepts such as Lebesgue integration and Lebesgue measure of sets. Today, I was telling my brother about how the irrationals on the interval [0,1] have measure 1, which is sort of counter-intuitive. Anyway, he proposed the following question. Let A...

It's a really tricky one to work out, but does anyone have any suggestions of how to measure compressive strength? Thanks a lot.

Hi In my shop, i have to design a fixture & want to calculate the force exerted by a moving element. Can anyone let me know the simpliest formula to calculate force ? Zirok

This is a practice final exam problem that has been giving me fits: Let E be a Lebesgue measurable subset of the interval [0,1] that has finite measure. Show that there exist two points x,y \in E such that x-y is rational.

I have teenaged daughters, and like many of their peers, they're obsessed with their body weight. Unfortunately, since I have access to accurate scales at my gym, I've never bothered buying a bathroom scale at home, and this is now becoming a source of friction. Rather than simply surrender...

Homework Statement Suppose that if k is a measurable function and {x : k(x) > 0} has a positive measure, then {x : k(x) >= p} has positive measure for some p > 0 Homework Equations Lebesgue measureThe Attempt at a Solution What is positive measure? is it just another word for measurable? Can I...

I would like to measure the distance between team players in the field. I need a device that can signalize when a distance between 2 players of the same team approaching a certain distance (2, 3 or 4 meters). The device must be relatively small (in order to have in on a players), to...

Dear all: I want to make a chamber as O2-free. The chamber is not perfectly sealed, so it has some small holes. I have used N2 gas (by pumping N2 from one input and escaping of O2 from holes in junctions). Now, I want to measure the amount of remaining O2 inside chamber. How can I do that? Is...

hi, i measured the emission spectrum of an LED with a monochromator connected to a PMT tube. the spectrum was measured at four different gain levels on the PMT. i want to check the PMT's linearity at the different gain level, so i want to compare the four data sets and check their...

Why is the measure and "finititude" of c critical? Why is it that c plays such a huge role in SR? Is it the actual relative measured speed of c, or is it that it's finite or both? I know that this may seem mirky. I "get" time dialation, length contraction, dopppler shift... What is the...

How can you measure the mean density of a neutron star if your only told its mass is three times that of the Sun?

I'm trying to measure solar radiation. Unfortunately I currently live in Bolivia, so I cannot get a solar cell. Instead I wanted to use a photodiode, but I wasn't able to find that either. Do you think that it is possible to measure solar radiation using a LDR? What about a phototransistor...

I just beginning to study measure theory. so far from what i understand so far , can we say in general, an integral is a measure, (ie it is nothing but a set function. a mapping F : \mathcal{F} \rightarrow \mathbb{R} where \mathcal{F} is a family of sets. does it make sense to say in...

Today my sister asked me to measure her height so she can calculate her BMI. Being the lazy lout that I am I told her to measure herself. And then I started thinking of ways she could actually measure herself. I am embarassed to admit I could only think of two ways: 1) Let's round out her...

I need to measure the capacitance of something as accurately and as quickly as possible. I also need the method to be very noise tolerant and not depend upon signal attenuation. I've been looking at several different methods so far on how to do this and so far I have found methods like the UTI...

I'm tinkering with a simple heat engine and I want to experimentally measure its work output. The engine pushes a piston once per test, and I know the work done is on the order of 1 kilojoule. The problem is that the engine does the work on the order of milliseconds, but depending on the...

i want to use measure the frequency of the square wave input at RA4/T0CKI of PIC16F84. so how should i start writing the code? anyone can provide some draft of the c code? thanks.

HI, THIS IS MY FIRST POST. I AM ACTUALLY A STUDENT AT A UK COLLEGE AND I AM STUDYING ELECTRONICS. I HAVE ALREADY PASSED MY COURSE WITH FLYING COLOURS WITHOUT SITTING MY FINAL. I AM NOW WORKING ON A PERSONAL PROJECT AT HOME. IM JUST HOPING THAT SOME OF YOU GUYS COULD LOOK AT THYE ABOVE CIRCUIT...

1. Suppose you are on the bank of a river at night. Opposite you on the other bank is a street light. You have a sensitive light meter and a meter stick. How wold you determine the distance across the river?

Hi I'm interested in measuring the recoil force of a weapon. Anyone got any ideas om how to go about doing this? I was thinking of using a load cell mounted to the weapon in the horizontal plane, but at the same time the weapon needs to operate more or less freely, which will be difficult...

I am trying to learn about Lebesgue measure. One of the questions I couldn't solve is this; Show that the following sets are Lebesgue measurable and determine their measure A = {x in [0,1) : the nth digit in the decimal expansion is equal to 7} B = {x in [0,1) : all but finitely many...

how to measure water level in a tank?? How to measure water level in tank in the cheapest way? The first thing that came to my mind is pressure sensor... but it's ruled out because it's expensive. Can somebody suggest a cheaper mechanism? And it is preferred to have a low power mechanism...