What is Measure: Definition and 999 Discussions

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.

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  1. S

    I Measure of existence? (Stanford Encyclopedia of Philosophy)

    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...
  2. H

    A How to measure the two-photon detuning in 3-level system?

    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...
  3. C

    B Is it possible to measure both position and momentum simultaneously?

    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...
  4. L

    Finding the measure of a set

    my question is how can I approch the problem ? And what is explicitly the set f(Eα)? {f(x) ∈ [a, b] such that what ??}
  5. J

    Best way to measure magnetic waves for electromagnetic induction

    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...
  6. S

    How to measure solar panel voltage using arduino?

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  7. AotrsCommander

    How do you measure time on a tide locked planet?

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  8. K

    I Measuring Time & Spatial Distances: Timelike vs Spacelike

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  9. K

    I Using position of free particle to measure time

    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...
  10. Math Amateur

    MHB Outer measure .... Axler, Result 2.8 ....

    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...
  11. SherLOCKed

    I Will the upcoming experiment PICO measure kSZ temperature anisotropy?

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  12. mef

    I How to Measure Speed of Light & Is It Constant?

    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...
  13. R

    A Measure of non-periodicity of almost periodic functions

    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...
  14. J

    Entropy: Does Disorder Really Measure Order?

    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...
  15. karush

    MHB 00.43 measure angle KPM

    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 $\angle{KPM}$ $s.\ 12^o \quad b.\ 28\dfrac{2}{7}^o \quad c. \ 42^o \quad d. \ 61\dfrac{1}{5}^o \quad e. \ 66^o$
  16. P

    I Standard measure of distance from equilibrium for all systems

    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...
  17. G

    Special relativity - measure of a rod and simultaneity

    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...
  18. V

    I How can we observe and measure interacting vacuum?

    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...
  19. S

    B What is dTau? The Role of dTau in Measuring Time on a Worldline

    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?
  20. M

    B Measuring Light Velocity Correctly: Thought Experiment

    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...
  21. BWV

    B Observers sending signals to measure expansion

    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...
  22. MathematicalPhysicist

    I Can you practically measure an infinite amount of some quantity?

    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?
  23. B

    Measure a varying force

    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...
  24. elcaro

    B Can we measure acceleration of galaxies and stars?

    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...
  25. C

    A Effects of a spatially nonuniform diffusion parameter

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  26. F

    MHB Proving Measure Space Properties of $(X,\bar{\mathcal{B}} ,\bar{\mu})$

    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 \{...
  27. K

    I How to measure a 10 Hz energy splitting of two energy levels

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  28. agnimusayoti

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  29. K

    I Smallest energy we can measure

    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...
  30. nick26

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    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...
  31. A

    Analysis Prerequisites Measure theory for ug student in physics

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  32. A

    Why does the top surface of a Silicon Die measure close to 0 ohms?

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  33. S

    I To measure a Planck length would require a black-hole photon?

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  34. S

    How to Measure Torque Required for motor to spin a rotating shaft?

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  35. sahilmm15

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  36. Heisenberg_white

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  37. K

    I Is it feasible to measure one way speed of light this way?

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  38. Mikkel

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  39. MathematicalPhysicist

    I A claim in measure theory which seems flawed to me

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  40. f9CSERS

    B Radian measure and real numbers

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  41. T

    How to Measure and Calculate Band Gap of a Photovoltaic Device

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  42. H

    A How can we measure these Hermitian operators?

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  43. A

    B How can we measure the wavelength of gamma-rays?

    How can we experimentally measure the wavelength of gamma-rays, say for about 0.7MeV? Can it be done without gamma-ray spectrometry?
  44. J

    How to Measure PCB Impedance and Component Coupling?

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  45. Jehannum

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  46. Math Amateur

    I Translation Invariance of Outer Measure .... Axler, Result 2.7 ....

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  47. Math Amateur

    MHB Translation Invariance of Outer Measure .... Axler, Result 2.7 ....

    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...
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