Measure Definition and 1000 Threads

Hi, I'm trying to show that Givien a probability triplet (\theta,F,P) with G\in F a sub sigma algebra E(E(X|G))=E(X) Now I want to use E(I_hE(X|G))=E(I_hX) for every h\in G since that's pretty much all I've for the definition of conditional expected value. I know this property should use the...

Hi, I've had an idea for a school physics project but I'm not sure if it will work. I want to measure the volume of water in a fixed shape container by passing an electrical current though it and monitoring how the resistance changes (with a potential divider circuit). My idea is that if I...

Homework Statement I know that temperature is a measure of the kinetic energy of gas molecules. at a given temperature different gases have the same average kinetic energy. But in solids and liquids there is also potential energy between the close atoms/molecules. how do i know that temperature...

Hi, I am working on a PCB that requires controlled trace impedance. I figured out width of the traces and layer stack thickness. How would I verify trace impedance once I get my PCB boards? What equipment do I need?

Hi All, I am measuring the impedance response of a piezoelectric component. It has a unique impedance response having a resonance portion (low impedance ~10-30ohmns) and large impedance (100k-200kohmns) within a 2kHz bandwidth. See the webpage for a figure showing the impedance behavior. The...

Hey! :o In $\mathbb{R}^d$ with the Lebesgue measure if $f \in L^p, 1 \leq p < +\infty$, and if for each $y$ we set $f_y(x)=f(x+y)$ then: $f_y \in L^p$ and $||f||_p=||f_y||_p$ $\lim_{y \rightarrow 0} ||f-f_y||_p=0$ Could you give me some hints how to show that?? (Wondering)

Could someone please explain why is chosen to measure ac sin/cos signals with radians instead of seconds? If ac waves is the behavior of a wave over time? Any feedback would be appreciated. Thank you!

Hi everyone, I have been trying to find an equation to measure the length of a coil. The coil is a metal sheet that wraps around the loops as it creates. It looks like a roll of tape. I have been using two different formula, but it is not giving me the right results. 1. L = 0.065449 (OD^2 -...

I have made a pitot tube that I am using to measure the velocity of air in a pipe. First, I placed the probe into the pipe such that the hole in the probe was parallel to the flow to obtain static pressure. It did this successfully and I proved this using a piezometer. However, I then turned the...

Hey! :o I want to show that when $\mu$ is a Borel measure in $\mathbb{R}$ with $\mu([0,1))=1$, which is a translation invariant, then it is also a Lebesgue measure. I have shown that $\mu([a,b))=b-a, \forall a,b \in \mathbb{Q}$. Is it enough to show that $\mu$ is a Lebesgue measure?? (Wondering)

Hi, When i Look at SI derived base units like Joule, Newton, pascal and base units like the Ampere which is defined as "The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre...

Hello, I have a one-dimensional path associated with timings, similar to: t0=0, x0=0 t1=0.1, x1=2 t2=0.2, x2=3 etc. Now I want to measure the velocity, acceleration and jerk at each time position. The first (and easy) way of doing this would be to loop through all points and compute...

Hey! :o Let $\mu$ be a Borel measure in $\mathbb{R}$ such that $\mu(I)\leq v^a(I)$ for each bounded interval $I$, where $a>1$. Show that $\mu=0$. Could you give some hints how to show this?? Do we maybe use the identity that for each rectangle R the outer measure of R is equal to the volume...

Hey! :o At any metric space, find a formula that gives the measure of the union of $n$ measurable sets, not necessary disjoint. If the sets are disjoint the measure of the union is $$\mu \left ( \cup_{n=1}^{\infty} A_n \right)=\sum_{n=1}^{\infty}\mu(A_n)$$ right?? And when the sets are not...

Hey! :o I have the following exercise.. Show that the $2$-dimensional Lebesgue measure of the graph of a continuous real function is zero. Could you give some hints what I could do?? (Wondering)

Hello, I would like to ask if its possible to apply pressure through fluid on a curved soft tissue such as silicon to measure the strain by using sensors (and what kind) such as shows in the attached image? If those were pressure sensors would they all show the same pressure or would the change...

Hey! :o I am looking at the proof that the outer Lebesgue measure that is defined by $$m^*(A)=inf\{\sum_{n=1}^{\infty}v(R_n):A \subset\cup_{n=1}^{\infty}R_n , ( \text{ where } R_n \text{ are open rectangles}) \}$$ is actually an outer measure. ($v(R_n)$, the volume of R is the product of the...

This might not be the right subforum, but I was told that measure theory is very important in probability theory, so I thought maybe it belonged here.I am confused about the difference between a measure (which is a function onto \mathbb{R} that satisfies the axioms listed here...

Is it possible practically to measure Hausdorff dimension of the surface of the Brain or the broccoli? For a broccoli Hausdorff dimension is equivalent to the so called box counting dimension, which is far more practical? I think, following the original definition Hausdorff dimension it is quite...

sorry for spamming the boards with all these questions lol- i promise this is my last post regarding n body problems. how do i measure error? i can't really find much on this subject. i want to use adaptive stepsize in my n body simulator I am guessing for the n-body problem i base my error on...

I know that gibbs free energy for say a body will be equal to = Gibbs free standard energy at 1M and Ph7(-rtlnkeq) (Where k is the concentration of product/ concentration of reactant at equilibrium)+rtlnk. How can we use the standard gibbs free for irreversible spontaneous processes? Is it...

Let's say that you are doing some Monte-Carlo simulations of a statistical system on a lattice and you observe scale invariance, meaning that you are at a conformal point. Can you get a numerical appreciation of the central charge? I know how the central charge is related on the free energy...

Hi, I have a High Voltage supply on an integrated circuit board installed in a Pentium I computer, that (in theory) can give a 0-3200 V Bias Supply to a Germanium detector, and only 0-1000V multimeters to test with. I have a Soviet analog multimeter, a cheap new one (digital) and...

In $\Omega = \{0,1\}^{\mathbb{Z}^{2}}$, consider the class $C$ of cylinders. Show that $C$ is algebra. At $w\in \Omega$, we call cluster point $x$ all points $z\in \mathbb{Z}^{2}$ that can be attained from $x$ by a path that only passes by open dots. In the $\sigma$-algebra generated by $C...

Hi all! Supposed I have my solution of sucrose, my light source, my polarizer. I shine my light source on my polarizer and plane polarized light passes through my solution, how then do I measure the how much my solution has rotated my polarized light by? And also, what kind of container...

Hi I need a formula that returns a value representative of the amount of ‘assortment’ a group shows. The groups are made up of individuals, all of a binary class (e.g. male or female), are of difference sizes, and can be from different populations (i.e. different ratio of males to females). I...

Hello, I recently started going through some lecture notes on linear systems and Fourier optics. (By the way, I just started with these, but so far the lecture notes are excellent. If anyone is looking to learn the subject but doesn't want to spend money on a textbook, the lecture notes, and...

Hi. I'm looking for a common electronic component that can be used to determine a temperature between room temperature and that of liquid nitrogen. -Multimetre probe failed, getting stuck at -135*C -Common NTC thermistors rapidly ascent past gigaohm range, even from 1Ω-5Ω initial values...

Let's say i spin an object around me with a greate velocity. At some point, i leave that object and it moves in a straight direction with a velocity of 0.9c. If so what was its velocity while it was revolving around me? How do i experes it?

Hi I am very confused how can measure voltage drop of a sheet if 1 Ampere is passed and the sheet has a resistance of 0,01 ohm in sq does it matter the thickness of the sheet ?

Why is it so important to the universe that if you measure the spin of an entangled pair and its up then the other particle must be down. It seems to have no practical use so why does the universe enforce this rule, how would reality differ if it wasn't true. And is it a law of symmetry...

Hi, My question is how do do laser interferometers measure distances of more than one wavelength? I know they rely on phase difference to calculate distances but phases only differ at 180 degrees at max so how do interferometers factor in going over 180 degrees in phase difference to...

Which equation is valid only when the angular measure is expressed in radians? a) α = Δθ / Δt b) ω= Δω / Δt c) ω^2 = ωo^2 + 2αθ d) ω = Vt/r (here T is a subscript) e) θ = 1/2αt^2 + ωαt Answer is D but why?? * I am totally lost so I can not show work

This question has came up to my mind and I think it really isn't easy as it sounds. In Newtonian physics we could use the Earth as the basic frame and compare velocities of cars, planes etc. on our planet. But in relativistic physics, things get complicated because, as we are in a...

If f=g a.e f and g are equal except at a measurable set with measure zero If two functions are not equal a.e what will then the negation be? Will there have to exist a set that is measurable, and f is not equal to g on this set, and this set has not measure 0? Or will the entire set...

1.Find the measure of <ACD for this one i thought I'd do 180-134-38 but 8 doesn't seem right to me?? 2.Find AC 3.Find AB 4.Find the area of the garden

Find the measure of angle $\angle BAC$.

Hey. Given that if you measure the energy of a wave function, the wave function must collapse to the eigenstate corresponding to the eigenvalue measured. Does that mean when you measure the energy of a wave function it must collapse the wave function into one of these stationary states...

Homework Statement Let (R,M,m) be Lebesgue measure space in R. Given E contained in R with m(E)>0 show that the set E-E defined by E-E:={x in R s.t. exists a, b in E with x= a-b } contains an interval centered at the origin Homework Equations try to prove by contradiction and use...

Homework Statement (a) Let \alpha:I=[a,b]→R^2 be a differentiable curve. Assume the parametrization is arc length. Show that for s_{1},s_{2}\in I, |\alpha(s_{1})-\alpha(s_{2})|≤|s_{1}-s_{2}| holds. (b) Use the previous part to show that given \epsilon >0 there are finitely many two...

Hello everyone, I am having a little difficulty understand precisely what Gibbs free energy is. I have read in my textbook that a negative change in Gibbs free energy implies that the substance under consideration will react/change spontaneously. As such, the more negative the Gibbs free...

I have read Max Tegmark's book "The Mathematical Universe" and he describes this thing called The Measure Problem as the biggest problems in physics. I am having difficulty understanding the problem so I will try to sum up my understanding of what he said. As a result of inflation, the volume...

This is an experiment I'll be undertaking for labs. Given the following equipment: - laser with modulation input - lenses and mirrors - function generator - high-speed photo detectors - oscilloscope and the setup as shown in the picture (the function generator is connected to the...

The context is that I am reading the proof that Lebesgue measure is rotation invariant Let X be a k-dimensional euclidean space. T is a linear map and its range is a subspace Y of lower dimension. I want to prove that m(Y) = 0 where m is the lebesgue measure in X. How to prove this...

The problem statement Let ##\mu## be a measure defined on the Borel sets of ##\mathbb R^n## such that ##\mu## is finite on the compact sets. Let ##\mathcal H## be the class of Borel sets ##E## such that: a)##\mu(E)=inf\{\mu(G), E \subset G\}##, where ##G## is open...

Lets say I have the following pipe with layers of different material (material is simply a example): As you can see in the middle there is a hole with just air thru it. If I fill that pipe with pressurized water, it will make OUTWARD pressure on that first steel pipe. Obviously because of...

Homework Statement . A space ##(X,\Sigma, \mu)## is a complete measure space if given ## Z \in \Sigma## such that ##\mu(Z)=0##, for every ##Y \subset Z##, we have ##Y \in \Sigma##. In this case, prove that a) If ##Z_1 \in \Sigma##, ##Z_1ΔZ_2 \in \Sigma## and ##\mu(Z_1ΔZ_2)=0##, then ##Z_2...

If \{ E_{k} \}_{k \in \mathbb{N}} is an increasing sequence of subsets of R^{p}, then: | \displaystyle \bigcup_{k=1}^{\infty} E_{k} |_{e} = \lim_{k \to \infty} |E_{k}|_{e} I proved: | \displaystyle \bigcup_{k=1}^{\infty} E_{k} |_{e} \geq \lim_{k \to \infty} |E_{k}|_{e} But I don't know how to...

"Measure of the same" From Newtons Principia: "THE QUANTITY OF MATTER IS THE MEASURE OF THE SAME, ARISING FROM ITS DENSITY AND BULK CONJUNCTLY." What does he mean when he says "is the measure of the same"? This phrase is used many times in his other definitions as well: "THE QUANTITY OF...

Hi, I've got a horizontally moving piston in a pneumatic cylinder, the piston is moving backward to suck the air into the cylinder from a hole at one end, this will lead to an air flow rate. I would like to measure the air flow rate at that instant. I was thinking of with a pressure sensor...