Measure Definition and 1000 Threads

Let (X, \mathcal{A}), (Y, \mathcal{B}) be measurable spaces. A function K: X \times \mathcal{B} \rightarrow [0, +\infty] is called a kernel from (X, \mathcal{A}) to (Y, \mathcal{B}) if i) for each x in X, the function B \mapsto K(x,B) is a measure on (Y, \mathcal{B}), and ii) for each B in...

Hi all! It's my first post, but I follow this forum since a couple of years. Now I have a big question for you: how to measure the CDM (or \LambdaCDM) power spectrum (better, the variance \propto k^3 P(k)) from the observations? What parameters are necessary?

I'm not completely familiar with measures yet, but am trying to be. I'm trying to show a that a few sets in Rn have measure zero and am having difficulty showing this using countable covers with a total volume less than \epsilon. Is there an easier way to show that a set has measure zero? If...

Can a wheatstone bridge be used to measure very high resitances with good accuracy? If not what problems interfere?

Question about measuring observables. If have 2 particle system the particle seperation Q=x1-x2 and total momentum P=p1+p2 are observables of the system as a whole and are commuting. How do you measure these observables. It would seem the only way to measure the separation is to measure the...

Hi guys, I'm new to the physics world and am getting a bit confused with some of the terms used, stopping me getting the maths i need. Essentially I need to measure the deflection at the end of a beam, supported at one end. Like this...

I have a brass rod with a 90 degree bend. I want to measure the outer and inner radii by placing a piece of string along both of them, and then straightening the string out to determine the linear measurement. Is this an accurate method?

Im currently enrolled in embedded systems (so bare with me, I am not too experienced) and we had to come up with a final project for ourselves. Well, I would greatly appreciate some help in understanding some concepts of DSP. I am working with the freescale DEMOACKIT with the coldfire...

Homework Statement Suppose X is a subset of R such that its complement has Lebesgue measure 0. Show that there exists a c such that for all integers n, c + n is in X. The attempt at a solution I've been thinking about this for a while and I just don't see how such a c could exists. Any tips?

Hi, Can someone please give me some suggestions on how to use a microwave oven to determine the specific heat capacity of tap water? This is not a homework question, but an assignment and I have absolutely no idea how I can do this.This experiment will be done at home, so please no immersion...

Hello; Homework Statement Let \mathcal{A} be the \sigma-algebra on \mathbb{R}^2 that consists of all unions of (possibly empty) collections of vertical lines. Find the completion of \mathcal{A} under the point mass concentrated at (0,0). Homework Equations 1st: Completion is defined as...

hello! I'm investigating on thermal conductivity of concrete blocks. I've got an equation for calculating K, which is Fourier's law. Q'=-k×A× ∆T/l I have to make the experiment simple as possible as i can. My plan is to heat one side of a block of concrete and measure the temperature difference...

This is a old one, but it is rare to find the general solution: Consider a string (a fuse) that, when burned from one extreme, takes one hour to burn, but irregularly (some pieces burn faster than another). Get two strings of this kind. The irregularities are different, you are only granted the...

Homework Statement A chord of a circle subtends an angle of θ radians at the centre of the circle. The area of the minor segment cut off by the chord is one eighth of the area of the circle. Prove that 4θ = π + 4 sin θ Homework Equations s = rθ area of sector = 1/2 r2θ area of minor...

Homework Statement A particle with charge q and mass m is tied by a (I assumed stiff) string of length L to a pivot point P, all of which lie on a horizontal plane. A uniform electric field E is placed over this system. If the initial position of the particle is at a point where the string is...

All the books I want to read about probability and statistical estimation require some understanding of measure theory. What is a good introductory text on measure theory you would recommend (assuming no prior knowledge of measure theory at all)? I want to be able to teach myself from the book...

Suppose one has a continuous complex valued function f:X->X, where X is some a non-empty compact subset of the complex numbers, and of the form Z_n+1 = f(Z_n; a1, a2,..., am), where a1, a2,..., am, with 0 <=|ai|<=1, for i =1, 2,...,m, are independant complex valued parameters, and n = 1,2,3 ...

I've seen the definition that an outer Lebesgue measure is defined as m_n^* (A) = \inf \left\{ \sum_{k=1}^{\infty} l(I_k) \, \left| \, A \subset \bigcup_{k=1}^{\infty} I_k \right} where Ik are n-dimensional intervals and l(Ik) is the geometric length. It is not actually clear to me if A has...

Hi, We're offered shaving mirrors by suppliers and sometimes their claims to the magnification are doubtful. I don't wish to upset any customers by giving incorrect information. So is there a practical way in which I can test the magnification myself? i.e. measuring the distance of the...

If you have two measurable sets A and B (not necessarily disjoint), is there an easy formula for the measure of the difference, m(A-B)?

I am trying to test different fuel types(for science fair, I am a sophomore in high school) and i was wondering if anyone new something that could be bought or made to measure the force of an explosion. My current ideas don't seem like they would be very accurate. Any help at all would be...

Or how did Coulomb measure quantity of electricity while doing his experiments that leads him to claim his famous Coulomb's law. F=k\frac{q_1 q_2}{r^2} You can measure F and r for sure. But how about q1 and q2 ?

Suppose that E has Lebesque measure 0. Prove that the set {x^2 : x€E} has Lebesque measure 0. Please help me. I have a problem which is unsolveable for me. Thanks!

Hi, Could someone give me some basic idea or information 1. How to measure the capacitance of interdigitated metal electrode of about 250 nm wide and 2 mm long on a silicon wafer, spacing between the electrode is 250 nm which is bonded bonded to copper pads. 2. Is it possible to measure the...

1. Basically this was a question directed by my subject lecturer. He needs me to describe a method on how to measure the amount of charges stored versus the spacing between the plates of a capacitor. 2. There are many types of ways and equations to measure the amount of charges stored but...

So I was trying to figure out how much power items at my house use. I have a standard looking power meter outside with the dial http://en.wikipedia.org/wiki/File:Electrical_meter.jpg" . I am trying to figure out though what unit the meter uses. My first instinct was watt hour, but I can't get...

Hi, I am pursuing graduate studies in economics, and I hear that "measure theory" is one of the classes that will impress admission commitees. I don't see anything by that name in my school's catalog. Does this class go by another name sometimes?

Dear all, I need a suggestion for my work. I have to measure the variation of some optical features of a material in presence and absence of a magnetic field. Hence, I need to “switch on” the magnetic field in a given instant time, to record the signal an then ““switch off”, or even to change...

Odd question that at first seemed to have an obvious answer: no. It all started when I realized that my friend is horrible at shuffling cards. They fall in large packets such that they fall in layers of unshuffled cards, rather than all of the cards being randomized. Then I realized to...

Some time ago, someone in this forum asked how you measure momentum. One of the answers said that if it's a charged particle, you can let it pass through a bubble chamber and a magnetic field, and measure the curvature of the bubble trail. But this isn't really a direct measurement of the...

I just encountered the Wikipedia page There is no infinite-dimensional Lebesgue measure, and I was left slightly confused by it. They say that a Lebesgue measure m_n on \mathbb{R}^n has the property that each point x\in\mathbb{R}^n has an open environment with non-zero finite measure, and then...

Hi everyone, I'm new to the forum and basically need help to build a little gadget to measure the intensity of light that passes through a tube filled with solutions of different dye concentrations. I have a 10530DAL photodiode, and a light source. What I'm doing now is measuring the...

Hello all, I am looking for some guidance on making a mechanism that will retract a 900 metre length of fishing wire by itself. The idea is that the wire can be pulled up to 900 metres from its source and once let go it will return to the spool like a tape measure (without a crank or...

Homework Statement I have been asked to design a device or assemble existing devices that will measure light intensity from a source across diifferent places on a table, the data has to be recorded electronically on a computer as well, the position of the measuring point has also to be...

This question came up recently, and I'm wondering whether or not it's true: Let (X,A,m) be a finite measure space. Let E_1,E_2,... be a sequence of measurable subsets of X of constant positive measure (i.e., there exists c>0 such that m(E_i) = c for all i). Then there exists a subsequence of...

I was wondering, are mass and energy specific to a particular reference frame, as well as things like time? Let's say for example, there's two observers, one in motion (A) and another stationary (B) standing along side my path of direction. If I were in accelerated motion, observer B would...

In the UK we live in a society which insists that all that matters is education and economic output, while the value of 'true love', which comes to us all in our teens, is widely denied. As a result, we are left with a 'broken' society, where 50% of relationships fail and where personal greed...

I've read from many sources that thermocouples or any other type of thermal transducer can't accurately measure the temperature of multiphase flows but none of them explain why. For instance, a mixture of air and ethanol going through a fuel line at 80 C. Why is it that if you stick a...

What theory are they? Set theory comes to mind but is that too broad?

Homework Statement Let (X_n) be a sequence of measurable subsets of \mathbb R such that \sum_{i=1}^\infty m(X_i) < \infty Define X = \bigcap_{i=1}^\infty \left( \bigcup_{j=i}^\infty X_j \right) Prove that m(X) = 0. Homework Equations Theorem. Let (E_n) be a sequence of...

Ok, here's the deal. I am trying to measure how much energy (best in watts, i guess?) a device uses over a seven hour period. I bought a device from lowes that measures watts and amps but how do I figure out [for example] if it says a device is using 8 watts and .2 amps right now, how much...

Homework Statement Let D be a class of sets of space X. There exists a unique smallest ring, R, containing D The Attempt at a Solution I'm working on the existence part. If D were finite, I think that it would be possible to use set differences to find a set, D', of smaller sets...

Hi: I am trying to review the way L^p spaces are treated differently in Royden. In Ch.6, he treats them under "Classical Banach Spaces", and then again, in his Ch.11 , under "Abstract Spaces". This is what I understand: (Please comment/correct) In the case of abstract...

What does it mean by \int f(w) P (dw) I don't really understand P (dw) here. Does it mean P (x: x \in B(x, \delta)) for infinitely small \delta ? For example, with P(x)=1/10 for x=1, 2, ..., 10 . How can we interpret this in term of the above integral Thanks...

1. Is an annual increase of lightning strikes worldwide a measure of global warming? 2. What has been the correlation between such discharges and temperature in recent decades?

Just for fun, I was thinking of trying to design one. How could we design a thermometer that could accurately measure negative temperature of a spin system?

My homework problem says to construct a thermometer which could accurately measure a negative temperature. I have no idea how to do this. Any help would be very appreciated!

Homework Statement Construct a sequence (G_n) of dense open subsets of \mathbb{R} such that \lambda \left( \bigcap_{n=1}^\infty G_n \right) = 0 and hence deduce that there exists an uncountable subset of \mathbb{R} of measure 0 Homework Equations \lambda is the Lebesgue...

Homework Statement An accelerometer—a device to measure acceleration—can be as simple as a small pendulum hanging in the cockpit. Suppose you are flying a small plane in a straight, horizontal line and your accelerometer hangs 12° behind the vertical, as shown in the figure. What is the...

Homework Statement To measure the acceleration due to gravity on a distant planet, an astronut hangs a 0.055 kg ball from the end of a wire. the wire has a length of 0.95 n and a linear density of 1.2*10 ^-14 kg/m. using electronic equipment, the astronaut measures the time for a transverse...