In mathematics, a measure on a set is a systematic way to assign a number, intuitively interpreted as its size, to some subsets of that set, called measurable sets. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the usual length, area, or volume to subsets of a Euclidean spaces, for which this be defined. For instance, the Lebesgue measure of an interval of real numbers is its usual length.
Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see § Definition, below). A measure must further be countably additive: if a 'large' subset can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets that are measurable, then the 'large' subset is measurable, and its measure is the sum (possibly infinite) of the measures of the "smaller" subsets.
In general, if one wants to associate a consistent size to all subsets of a given set, while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a σ-algebra. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.
Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure. Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system.
I'm reading the article on the Many Worlds Interpretation in the Stanford Encyclopedia of Philosophy. I'm keeping up well, but this excerpt uses things I'm very unfamiliar with:
I guess some characters weren't recognized. It's Section 3.6 here. I'm somewhat familiar with Wigner's Friend, but...
hi,everyone!
if i have a 3-level system,like this：
now,i want know how to measure the two-photon detuning in this system,and i‘ve known that,Δ2 and Δ3 are the two- and on-photon detunings,like this paper said:
my questions are below:
1.how other researchers measure these two parameters?
they...
A simultaneous measurement of both a particle's position and momentum may be successfully accomplished if more than one photon were utilized for the measurement. A non-demolishing measurement is possible if the emitters were aligned such that each would offset the other’s recoil of the target...
Hi all,
Looking to measure some magnetic waves being generated at an electric coil. Freq is between 0-20kHz and magnitude is pretty small <1T. Any have suggestions for the best tool to measure and log data of this magnetic waveform?
Googling around, I found meters like this: [Possible spam...
I would like to make the solar system a bit smarter. The challenge is that the voltage is high (up to 351Vdc) and the solar negative is typically 40-70V UNDER the "system ground". "System ground" is shared among battery negative, charger negative, inverter ground and ground cable digged down...
This is both a biomechanical and sociological question.
There is a planet. It orbits around a large star whose goldilocks zone is significantly far from the star that the orbit is measured in thousands of Earth-years1. The planet is tide-locked2. The planet is otherwise the same size and...
My book writes
" The geometric distinction between timelike and spacelike distances is mirrored in the devices used to measure them. A clock is a device that measures timelike distances; a ruler is a device for measuring spacelike ones. Two nearby points or a timelike world line are timelike...
Hartle, Gravity
"An observer in an inertial frame can discover a parameter ##t##with
respect to which the positions of all free particles are changing at constant rates.
This is time"
Then goes on to say
"Indeed, inertial frames
could be defined as Cartesian reference frames for which Newton’s...
I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ...
I need help with the proof of Result 2.8 ...
Result 2.8 and its proof read as follows
In the above text from Axler we read:"The doubly indexed collection of open...
I know for sure PICO will be measuring polarization anisotropies with high fidelity. In addition, the PICO science paper shows that it will make full-sky Compton-y maps but the plots are mostly limited to l=1000. Will PICO be able to measure kSZ temperature anisotropy at l=3000?
The question constantly arises how the speed of light is measured and what does it mean that the speed is constant, including at remote points for the observer, including at points beyond the local frame of reference, as you understand it in general relativity (GR).
First of all, it should be...
As is well known, almost periodic functions can be represented as a Fourier series with incommensurable (non-multiple) frequencies https://en.wikipedia.org/wiki/Almost_periodic_function. It seems to me that I came up with an integral criterion for the degree of non-periodicity. The integral of a...
Let's say you have a very dirty small room room and a giant clean library (lots of organized books) and let's say these occupy the same number of microstates. The entropy according to this equation is the same for the library and the room. But one is more ordered than the other one. How does it...
Ray $\overline{PK}$ bisects and the measure of $\angle{LPM}$ is $11x^o$ and the measure of $\angle{LPK}$ is $(4x+18)^o$
What is the measure of
$\angle{KPM}$
$s.\ 12^o \quad b.\ 28\dfrac{2}{7}^o \quad c. \ 42^o \quad d. \ 61\dfrac{1}{5}^o \quad e. \ 66^o$
Is there a standard way to measure how far a system is displaced from equilibrium that can be applied to all physical systems? So, for example, a ball that is kicked, a spring that is stretched, a liquid that’s heated, and a charged battery are all systems that are displaced from equilibrium. I...
Hi, I´m trying to solve a special relativity problem, and I think I need some help. There are two inertial frames of reference, ##O## and ##O'##, the last one moving with relative velocity ##v## in the ##x## direction. There's a rod with length ##L'## fixed to frame ##O'##, such that front end...
When studying the QFT, one considers the vacuum state when the field is not excited and therefore no particles are present.
Now for the matter fields this makes sense to me. But what about the radiation field? Suppose we have an arbitrary small volume of space in the universe without any matter...
I heard dTau measures time for the person traveling on a worldline. If the person traveling on that world line chalked marks on the world line every 1 minute, would those intervals be the same distance from each other?
Let there be a track 450,000 km long and a rocket 300,000 km long with a laser attached to the bottom of it's back end with a clock beside it, and a second synchronized clock attached to bottom of its front end. Both clocks were also synchronized with a track clock while the rocket was parked...
Googling you find a current expansion rate of ~70 km/s or 0.0002C but of course we don't observe this with objects in the solar system as local gravity prevents this expansion - otherwise a distances would increase by a light second roughly every 71 minutes (maybe I’m missing some here...
Suppose we measure some speed or energy of something with a suitable device or instrument. Now suppose the quantity that is being measured exceeds the capabalities of the measuring device either from above or below. How can we know if this quantity is indeed finite and not infinite?
how do I measure a rapidly and cyclically varying vertical force
I have a rotating mass that generates centrifugal and aerodynamic forces. I want to see the variation thru one rotation at steady state. how do I measure the vertical force generated and display it on my computer screen, with a...
As for example we see a large void, the Great Repeller, which in fact is an underdense region, and with respect to this region, matter seems to be repelled by this region. The explenation for that is that matter outside that regions pulls on the matter inside it. But if that is really the...
Einstein famously derived his relation between the diffusion constant of Brownian motion, particle mobility in a disippative medium, and temperature by considering Brownian motion in a harmonic oscillator potential. The result, $D = \mu k_BT$, is derived assuming that the mobility $\mu$ is...
Hello, I have problems with this exercise
Let $(X,\mathcal{B} , \mu)$ a measurement space, consider
$\bar{\mathcal{B}} = \{ A \subseteq{X} \; : \; A\cap{B} \in \mathcal{B}$ for all that satisfies $\mu(B) < \infty \}$, and
for $A \in \bar{\mathcal{B}}$ define
$\bar{\mu}(A) = \left \{...
Hello! If I have 2 energy levels split by something of the order of 10 Hz (they can be connected by an electric dipole moment i.e. ##\Delta J = 0## and they have different parities), what would be the best way to measure this difference (even 10% error would be good, but the lower the error the...
I assume that air have ##1 kg/m^3## density.
Therefore, using Bernoulli equation, on upside and downside of my test object, there is a differential pressure ##\Delta P##:
$$\Delta P=0.5*(v_2^2 - v_1^2)$$
From cases:
(a) ##v_1 = 1 m/s## and ##v_2 = 2 m/s##, then ##\Delta P = 1,5 Pa##
(b) ##v_1 =...
Hello! Assuming we can bring 2 energy levels very close to each other (e.g. by applying a magnetic field), what is the practical limit (in terms of lab equipment) on the smallest energy difference that we can measure? And what is the relative error on it, that can be obtained? For example if the...
Hi,
I need to come up with a math model for a digital ignition system. I've been thinking about it and I think that I need to measure 2 things to be able to calculate when I have to start charging the coil. They are the angular velocity and the acceleration but how can I do it? the idea is to...
Hi,
I would like to know if an undergraduate student in physics could be able to study measure theory in order to have a better understanding of the probability theory and go further in this way (stochastic process) ?
Assuming a first year of calculus and the level of "Mathematical methods in...
I have gotten access to a large bare silicon die (almost 1" across, 14nm process) that my company gets from a fab. I've been monkeying around with an ohmmeter placed at the top of the die, with the test leads placed at various points around the die, and I almost always measure the same value...
I was viewing this video in which the narrator says that the energy of a photon that could discern a Planck length would require a photon of such high energy that it would be a de factor black hole. Is this accurate?
I've harvested a motor from a cordless drill and connected it to a belt which turns a rotating shaft. The motor pulley and the pulley on the other side of the belt are roughly the same size, which a fairly small radius (5 mm maybe?).
The issue I'm running into, which I don't fully understand...
I am not getting any ideas to solve this without the universal constants. The method that I want to use invloves the speed of light, which is a universal constant.
Theoretical experiment for measuring one way speed of light
From 1905 to this day we have not experimentally measured the one way speed of light between a source to the detector only the roundtrip from the source to the detector and back again. We just assume that the speed of light is the...
Hello, I'm struggling with this question
A star is observed close to the center of the Milky Way and from its spectrum we find that it is a type A3 star. Its observed magnitude is m_v = 25. There is only a diffusive gas between us and the star, so we can assume an extinction, of 1 magnitude per...
The claim states the following:
Let ##(X,\mathcal{A},\mu)## be a measurable space, ##E## is a measurable subset of ##X## and ##f## is a measurable bounded function which has a bounded support in ##E##.
Prove that: if ##f\ge 0## almost everywhere in ##E##, then for each measurable subset...
Formula used : arc length = radius × angle (in radian).
I interpreted this as:
•Taking a unit circle, we get "angle (in radian) = arc length".
This means radian measure of an angle is arc length, which can be represented on a real number line. Hence, it is a real number.
Is this way to...
Hi there!
I bought a photovoltaic cell. I would like to learn its band gap at 300 K and 0 K. How can i measure band gap or calculate? I have already measured Temperature to Vmax values.
I couldn't find any good information about this topic.
Thanks.
Hi Pf,
I am reading this article about generalization of Pauli matrices
https://en.wikipedia.org/wiki/Generalizations_of_Pauli_matrices#Generalized_Gell-Mann_matrices_%28Hermitian%29
When i receive a qubit in a given density matrix , i can measure the mean values of the Pauli matrices by...
I am designing a PCB with multiple capacitors on it. I would like to measure the board impedance to compare it with results from Ansys Q3D. I can connect it to an impedance analyzer and easily get the impedance. I have recently thought about other meaningful measurements that I may be able to...
Ideally, the method should be accurate down to 0.01 millimetres or better.
We're probably talking pipes of up to 150 mm (6") diameter.
Accurately measuring the actual diameter of the pipe is of less importance - it's how much it expands that matters.
My idea is wrapping something around the...
I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ...
I need help with the proof of Result 2.7 ...
Result 2.7 and its proof read as follows:
In the above proof by Axler we read the following:
" ... ... Thus
... ##\mid t +...
I am reading Sheldon Axler's book: Measure, Integration & Real Analysis ... and I am focused on Chapter 2: Measures ...
I need help with the proof of Result 2.7 ...
Result 2.7 and its proof read as follows:
In the above proof by Axler we read the following:
" ... ... Thus
... $\mid t + A...