Measure Definition and 1000 Threads

I am trying to prove if f and g are Riemann integrable, then fg is also Riemann integrable using Lebesgue's integrability criterion. I already proved that a Riemann integrable function is bounded. Not much harder to show fg is too bounded. How do I show that [a,b] is of measure zero? I can't...

For a 2D vector field {F}=P(x,y)\vec{i}+Q(x,y)\vec{j} curl {F} = \frac{\partial Q}{\partial x}+\frac{\partial P}{\partial y}\vec{k} So that's the rate of change of the j component of a field vector with respect to x plus the rate of change of the i component with respect to y...how does...

Do you think having Bogachev's Measure Theory (vol. I) as a first exposure to measure theory sounds a good idea? I mean while I can understand well the concepts presented in the book, I find some techniques used in the proof section quite hard to follow. :confused:

Could we use the fact that all countable sets have zero measure to prove that their must be a larger infinity. we know countable sets have measure zero because I could just start by making boxes around each number and then add up their widths. For the first number I will make a box...

Entropy is a measure of energy availiable for work ? "Entropy is a measure of energy availiable for work". Can someone explain this to me? Give some examples that show in what sense it is true. It has to come with a lot of caveats, proviso's etc. because its simply not true on its face. I...

I know the textbooks say light travels 186,282.4 mps, but if time slows for a very fast object, how do scientists really know the speed of a photon?

Homework Statement Prove (using Lebesgue outer measure) that $$m_{2}^{*}(\{(x,y)\in\mathbb{R}^{2}\colon x>1,0<y<x^{-2}\})=m_{2}^{*}(A)<\infty$$ The Attempt at a Solution I'm not sure if this is valid proof but I'd have done it like this: $$\int_{1}^{k}\left|x^{-2}\right|\, dx <\infty \forall...

[SOLVED] how to build a device to measure in angstroms Hello, I was looking at a micrometer and wondered how one could build a similar device for meauring in angstroms. Could someone point to some theory that discusses the same ( for example if i have to forge a screw what would be its...

What is a measure of "jointness" of a joint distribution? Correlation?

Hello, I have a particular derivation of a four-vector integration measure, basically changing the measure to some related more useful measure - but I'd like to do this in 3-vector notation. Here it is, from the integral...

A friend asks me this. If considering the equation: ∫\frac{dQ}{T}, then it is technically feasible to work out some forms of expressions with measurable physical quantities like temperature and specific heat, therefore it is possible to work out a precise value for entropy change. But is there a...

Let's say I simulate a simple filter in simulink and I want to see the transfer function graph (bode plot) of it. What do I do to get Vout/ Vin? I tried using the "PS Divide" Block and divide the output voltage and the input voltage, but that didn't work. Could anyone help me out here? This...

It is well-known that the velocity of an object can only be determined in relation to the velocity of another object (the two trains in a station). Einstein's relativity theory limits the velocity of an object to the speed of light; it also been demonstrated that no matter what the velocity of...

How can you measure a capacitance of an isolated object, like a conductive sphere or a coil? Can it be done using a typical RLC meter? I've tried touching only one end of the probe to the object in question, and that seems to give sporadic readings, but within the order of magnitude of the...

Thermodynamics--Gibbs free energy: what can we actually measure in the lab? Hi everyone: I am getting back into thermodynamics after a long absence and have realized that there are basics I have never understood. If a patient person could either explain or direct me to an online resource...

In a simple circuit with a power source, electric conductors and a load, you can measure the power being dissipated by measuring the voltage across the load and multiplying by the current. If you only have access to one conductor, for example, an electrical distribution cable, how can you...

I had a question about about the integration measure for the path integral after a unitary change of variables. First they consider a 4D spacetime lattice with volume L^4. The measure is \mathcal{D}\phi = \prod_i d\phi(x_i) They expand the field variables in a Fourier series...

How does the Cut-Back method work experimentally? The Cut-Back Method is a method to determine the propagation losses of fibers or waveguides. I have found tons of articles stating they used this method in comparison but no detailed information on how one can execute it successfully...

Hi all, Sorry if this is in the wrong place. I'm trying to understand probability theory a bit more rigorously and so am coming up against things like lebesgue integration and measure theory etc and have a couple of points I haven't quite got my head around. So starting from the basics...

Measure of the "Sharpness of a curve" I have a set of curves that belong to the family of curves y=\frac{c}{x^m}, where m and c are parameters. The attached picture (save.png) shows three such curves for different values of m and c. Now these curves have different 'sharpenss' of curvature (to...

Homework Statement So I have before me a set of springs about this size: http://tinyurl.com/8g7kvu5 . Some are a little longer, some are wider, and some are slightly stronger. I need a home-made way to measure how much potential energy each spring can build up if I compress it at 2 inches or 3...

I have always struggled in understanding probability theory, but since coming across the measure theoretic approach it seems so much simpler to grasp. I want to verify I have a couple basic things.So say we have a set χ. Together with a σ-algebra κ on χ, we can call (χ,κ) a measurable space...

So I know that the Lebesgue outer measure of a set of only countably many points is 0. An example of this is the rationals as a subset of the reals.I want to make sure my intuition behind this is correct. The process: Now, if we are going to take the Lebesgue outer measure of the rationals, we...

I'm trying to determine the best method of measuring impedance from a wire through it's glass coating to determine if the thickness of the coating is correct. At the moment I do not have the actual frequency, I just know it is in the RF range and that my impedance needs to be 50 ohms. I've...

HI ALL We have a big building.there is a ring around the building as earthing system. we put bare cooper wire in trench around building . NOW,we are going to measure the resistance of earthing system with Earth tester. Our device is MI 2124 ,one production of METREL . how I can...

Homework Statement Let ( \mathbb{R}^k , \mathcal{A} , m_{k} ) be a Lebesgue measurable space, i.e., m_{k}=m is a Lebesgue measure. Let f: \mathbb{R^k} \to \mathbb{R} be a m-integrable function. Define a function \mu : \mathcal{A} \to [0,\infty] by $$ \mu(A) := \int_{A} f(x) dx $$ with A \in...

I have used two different Gauss meters to measure the same NdFeB rare Earth N45 (45 MGOe) disc magnet (D28X6mm), with a Residual Induction (Br) of 1320-1380 mT. The first is a Dr. Steingroever GmbH FH51 Gauss meter. It would take readings of up to approximately 1.2 Tesla (12,000 Gauss)...

Homework Statement I was wondering, how can i measure the specific heat capacity of a solution? Not the exercise, just doing an experimental work to measure it. Homework Equations q=mct The Attempt at a Solution I'm lost.

So right now I am shadowing a 8th grade science teacher at Drew Charter school and I was helping a student work on his idea come up with a science fair project. The student wants to study what kind of smells do dog's prefer over other's but using household (safe items), food, and dirty...

So I'm beginning to study photonics and the issue of excessively accurate reference times for clock signals are mentioned. I understand that natural frequencies are often compared to determine the accuracy of a time measure (olden times it was vs astronomical events and nowadays it's against...

I'd like to know with some precision how the position of a particle is measured. Just to focus the potential answers, I'd like to know some details of it because perhaps with that I could get a better understanding of some basic principles such us: 1) commutation relation of x and p 2) some...

I've always been confused by this concept: How do you measure center-to-center in engineering terms when the objects are on different x and y coords? When objects cross the same x or y plane it's simple enough, just draw a straight line from center to center. When they aren't on the same x or y...

hello! which is the principle and technology that enables us to measure objectively myopia, astigmatism and presbyopia? thanks!

Hi all, I've got 2 bevel gears that I'm recreating in CAD, I've got the shapes down it's just a little bit of fine tuning with the dimensions, for which I need the module/diametral pitch of the gears. The pinion has 9 teeth and a tip diameter of roughly 39.3mm, and the gear has 14 teeth and a...

The basic concepts used in physics are mostly derived from everyday human experience. Hardly surprising for a subject which usefully describes our contingent physical circumstances. For example dynamics involves ordinary and familiar concepts like mass, and changing distance is described as...

I have attached part of 2 pages from Justin T. Moores dissertation. I am wondering why he says every set of size \aleph_1 has measure zero. He is probably using some axioms that i am not familiar with. And I am not sure what the k_2 is. He says this towards the bottom of the...

Plenty of Dark Matter Near the Sun http://www.sciencedaily.com/releases/2012/08/120809090423.htm Evidence for dark matter near the sun "We are 99% confident that there is dark matter near the Sun," says the lead author Silvia Garbari. In fact, if anything, the authors' favored dark...

I had an experiment in my mind and I wonder if anyone could help me out with this, I'm just familiar with the basic concepts of relativity so all help is appreciated. Imagine the following case below in the image I have posted (it's possible this image will be gone as time moves forward)...

why do we use light signals to measure time? In flat spacetime Let's say we have two observers in one's rest frame the other one has velocity v. We suppose each of them sent light signals to other one with time separation of Δt(their proper time) and measure time accordingly.we see time...

We all know what it means to be homogeneous in a "hand waving" sort of way. And, of course, there are abstract mathematical definitions for a homogeneous space. I have been unable to find a physical measure of homogeneity which could be applied to a ensemble of particles, box of rocks, or the...

I wasn't quite sure where to post this question, so please forgive me if I chose the wrong place. Essentially I'm looking for an explicit expression for the Haar measure on SO(4), i.e. in terms of angles, or if you prefer, expressed in terms of the Lebesgue integral over a subset of the Lie...

Hey folks, some weeks ago we had an trial-exam and one of the questions there was: "How do you measure phase and group velocity." That question really got me. Having a fastly oscillating wave as carrier and then the envelope. Can someone help me? Is it impossible for the phase, as it is...

Hello everyone, I needed to know more about measure theory so I'm reading in Rudin's Principle's of Mathematical Analysis, somewhere in the chapter, he says: We let E denote the family of all elementary subsets of $R^p$... E is a ring, but not a $\sigma$-ring. According to my understanding of...

I realize that this would probably be patently obvious if I understood the nature of particles, but how is it possible to measure the momentum of a particle with just one measurement? Doesn't determining velocity require multiple measurements?

You have cement block placed on Earth and moon with same mass. When you try to push the block it would be easier to push the block on moon than on Earth i.e. the inertial property shown by block on Earth is more than it shows on moon. But the mass is same on Earth and moon. When you have inertia...

hello! lets say we have a long tube with variable diameter and with variable direction is it possible to measure the diameter of its walls in each point of its lenght, by sending waves or something from its edge towards inside it ? imagine it is not possible to access the outside of the...

suppose an electrostatic sphere and an very sensitive magnetometer are placed in a space that has magnetic field outside been screened ideally. Can we know the speed of the Earth transverse in the universe by dealing with the data obtained?

I'm asked to get the noise signal from the constant current source I built with a 741 Op-Amp. I know I have to read it from the scope, but how do I go about the connections to the scope, and what other procedures must I follow to read the noise contribution? Pls refer to the figure provided in...

Dear all, I have a question about the polarization measurement. In the photo production reaction like γp→πp with polarized photon beam of energy GeV. how to measure the polarized components of proton. A paper said that the focal plane polarimeter, consisting of two 55 cm CH2 analyzer...

I was thinking about (for experience, since somebody's probably already done it) making a sort of generator, with the shaft of the turbine connected to four perpendicular flat pieces. In water, waves would smash against the pieces, rotating the shaft and hopefully generating electricity. The...