# 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. ### 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. ### 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. ### 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. ### 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. ### 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...

7. ### How to measure solar panel voltage using arduino?

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...
8. ### How do you measure time on a tide locked planet?

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...
9. ### I Measuring Time & Spatial Distances: Timelike vs Spacelike

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...
10. ### 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...
11. ### 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...

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13. ### I Will the upcoming experiment PICO measure kSZ temperature anisotropy?

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?
14. ### 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...
15. ### 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...
16. ### 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...
17. ### 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 $\angle{KPM}$ $s.\ 12^o \quad b.\ 28\dfrac{2}{7}^o \quad c. \ 42^o \quad d. \ 61\dfrac{1}{5}^o \quad e. \ 66^o$
18. ### 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...
19. ### 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...
20. ### 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...
21. ### 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?
22. ### 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...
23. ### 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...
24. ### 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?
25. ### 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...
26. ### 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...
27. ### A Effects of a spatially nonuniform diffusion parameter

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...