What is Measurable: Definition and 130 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. nomadreid

    I Measurable cardinal to get inner model, also in cumulative hierarchy

    If κ is a(n inaccessible) measurable cardinal, then there exists an elementary embedding j:(V,∈)→(M,∈), with critical point κ, whereby (M,∈) is an inner model of ZFC and the construction of j can follow through taking a κ-complete, non-principal ultrafilter U and constructing κV/U. In the von...
  2. Math Amateur

    MHB Measurable Functions .... Lindstrom, Proposition 7.3.7 .... ....

    I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Proposition 7.3.7 ... Proposition 7.3.7 and its proof read as follows: In the above proof by Lindstrom we read the...
  3. Math Amateur

    I Measurable Functions .... Lindstrom, Proposition 7.3.7 .... ....

    I am reading Tom L. Lindstrom's book: Spaces: An Introduction to Real Analysis ... and I am focused on Chapter 7: Measure and Integration ... I need help with the proof of Proposition 7.3.7 ... Proposition 7.3.7 and its proof read as follows: In the above proof by Lindstrom we read the...
  4. Michael Price

    A Measurements and electroweak gauge invariance/transformations

    Most gauge transformations in the standard model are easy to see are measurement invariant. Coordinate transformations, SU(3) quark colours, U(1) phase rotations for charged particles all result in no measurable changes. But how does this work for SU(2) rotations in electroweak theory, where...
  5. Kevin Chieppo

    B Exploring QM: Is Uncertainty a Physical or Measurable Limitation?

    I'm a hobbyist physicist and I just started studying QM through watching Leonard Susskind's lectures on the Stanford Youtube channel. I get the idea of it being impossible to precisely know both a subatomic particle's position and momentum, but is this actually a physical limitation? Or is it...
  6. CalcNerd

    I Measuring Charge & Angular Momentum in Black Holes

    It is stated that a Black hole has only mass, angular momentum and charge for properties, but since it is black ie no light escapes its event horizon and charge E/M is related (same speed) as light (photons), how can unbalanced charge be detected? And since many Black holes have electron...
  7. A

    Need a machine to impart measurable charge (Coulombs) at will

    Hi there, Does anyone know of a machine/equipment/technology, etc. that allows a person to give charges (Coulombs) to conductors in a measurable way? I've rubbed a ton of plastic rods with fur but that's not too measurable. The Van De Graaff is overkill or it if I try to connect alligator clips...
  8. J

    MHB Measure Theory - Existence of Fsigma set contained in measurable set

    Problem: Let $E$ have finite outer measure. Show that $E$ is measurable if and only if there is a $F_\sigma$ set $F \subset E$ with $m^*\left(F\right)=m^*\left(E\right)$. Proof: "$\leftarrow$" To Show: $E=K\cup N$ where $K$ is $F_\sigma$ and $m^*(N)=m(N)=0$. By assumption, $\exists F$, and...
  9. M

    A How to prove that cuboids are Lebesgue measurable?

    Hello, how do I have to start to prove that cuboids are measurable in the context of the Lebesgue measure? Best wishes Maxi
  10. C

    Measurable consequences of entropy of mixing

    Most textbooks include an example of entropy of mixing that involves removing a partition between two (in principle) distinguishable gases, and compare this to the case where the two gases are indistinguishable. What I’ve not yet been able to figure out is what the consequences of this...
  11. B

    Sup. and Lim. Sup. are Measurable Functions

    Homework Statement For a sequence ##\{f_n\}## of measurable functions with common domain ##E##, show that the following functions are measurable: ##\inf \{f_n\}##, ##\sup \{f_n\}##, ##\lim \inf \{f_n\}##, and ##\lim \sup \{f_n\}## Homework EquationsThe Attempt at a Solution It suffices to...
  12. B

    Composition of a Continuous and Measurable Function

    Homework Statement Suppose that ##f## and ##g## are real-valued functions defined on all of ##\Bbb{R}##,##f## is measurable, and ##g## is continuous. Is the composition ##f \circ g## necessarily measurable? Homework EquationsThe Attempt at a Solution Let ##c \in \Bbb{R}## be arbitrary. Then...
  13. B

    Theorem About Measurable Sets

    Homework Statement I am working through a theorem on necessary and sufficient conditions for a set to be measurable and came across the following claim used in the proof: Let ##E## be measurable and ##m^*(E) = \infty##. Then ##E## can be written as a disjoint union of a countable collection of...
  14. Paul Colby

    I Are Evanescent Gravitational Waves Measurable

    Hi, (all discussions here are in the extreme weak field approximation about Minkowski space) For the last couple of years I've been looking into the production and reception of radio frequency gravitational waves. It's kind of a retirement project the main goal of which is to get a better...
  15. T

    I Help with measurable cardinal

    Please help me, I am an idiot ) From here: https://en.wikipedia.org/wiki/Measurable_cardinal measurable cardinal is a certain kind of large cardinal number. In order to define the concept, one introduces a two-valued measure on a cardinal κ, or more generally on any set. For a cardinal κ, it...
  16. D

    Show a limited function is measurable

    Not sure about the translated term limited (from German); perhaps cut-off function? Homework Statement Let f be a measurable function in a measure space (\Omega, \mathcal{F}, \mu) and C>0. Show that the following function is measurable: f_C(x) = \left\{ \begin{array}{ll} f(x) & \mbox{if }...
  17. R

    Out of phase light/photons, would it be measurable?

    With sound you can create out of phase signals and they cancel out, this is not possible with light as it does not interact like sound does. If it was possible to create two light sources in that were exactly 180 degrees out of phase with each other and aim them at a common point (eg an...
  18. C

    Countable union of Jordan sets is not always Jordan measurable

    Homework Statement Show that the countable union or countable intersection of Jordan measurable sets need not be Jordan measurable, even when bounded. The Attempt at a Solution For countable intersection, I think the rationals from 0 to 1 will work, each rational have jordan measure zero...
  19. NihalRi

    When is resistance large enough to be measurable?

    Homework Statement We conducted experiments using copper wires to observe the effect length has on resistance. We measured lengths from 5 - 30 cm and used a multimeter to measure the voltage while supplying a constant current from a power supply package. The multimeter measures in mV. Plotting...
  20. D

    B Can the space (or else measurable) be actually infinite?

    The (most popular) flat model of Universe is space-infinite. How the infinity is measured? Can you give me references to the papers about the actual infinity of space?
  21. Nick tringali

    How can you ensure you pull a mass at a constant velocity

    In an experiment I hope to carry out, one of my "constants" are velocity (constant velocity). I must say a way I will keep this value constant. For example I can not say "I will try my best to pull mass at a constant velocity" "I will take a video of the experiment. When I examine the video, I...
  22. Oleg Melnichuk

    Intro Physics Which book written: only measurable quantity - length?

    In which book is it written that the only measurable physical quantity is the length? Task. In any book I've seen thoughts and images that very clearly illustrate the sequence of actions and implicit assumptions when measuring something. The result has been that in any measurement key was the...
  23. S

    Why is impact parameter not directly measurable?

    I'm trying to understand a few things about the kinematics of collision processes. I guess it's because we calculate the scattered angle of the projectile and then back calculate to get a value for the impact parameter. Is this right?
  24. newjerseyrunner

    Over what scale is curvature measurable

    Imagine I have three space probes that I send out radially. They have a superluminal way to determine each other's relative position to each other instantaneously. If each one measures the relative position of the other two and comes up with an angle for them, how far away would they have to...
  25. Dr Wu

    In Need of Some (Measurable) Illumination

    < Mentor Note -- thread moved to Astro forum from the Sci-Fi Fantasy forum > I apologise in advance if this comes across as hopelessly esoteric, but here goes: picture a 250 metre diameter sphere suspended in space (okay, in orbit round the Sun) with a surface temperature of 8,000 K. Now I'm...
  26. C

    How is enthelpy not directily measurable?

    I keep hearing that enthalpy is not directly measurable and that on it's own it carries no physical signifigance. But if you have a gas in a container for example, it has some internal energy which I'm assuming is measurable (a least in principle), and you can also measure its pressure as well...
  27. J

    Real analysis exam problem

    Hi, I was leafing through some old exams of our Real analysis course, and I found this puzzling problem: "Let A⊂ℝ be Lebesgue-measurable so that for all a∈A, i = 1,2, ... (1) m1( {x∈ℝ | a+(3/4)i-2 < x < a + i-2} ) < i-3 Claim: m1(A) = 0." Initially I thought this may have something to do...
  28. M

    MHB Show that it is metric and the measurable is 0

    Hey! :o In a space of finite measure, if $f$ and $g$ are measurable we set $\rho (f,g)=\int \frac{|f-g|}{1+|f-g|}d \mu$. Show that $\rho$ is metric and that $f_n \rightarrow f$ as for $\rho$ if and only if $\forall c>0$ we have that $\mu(\{|f_n-f|>c\})\rightarrow 0$.What does "$f_n \rightarrow...
  29. K

    MHB What is this theorem about measurable functions saying?

    **Theorem:** Let $(\Omega,\mathcal{F})$ be a measurable space and let $f:\Omega \rightarrow Y$ be a given function. Let $\mathcal{A}$ be a collection of subsets of $Y$. If $f^{-1}(A) \in \mathcal{F}$ for every $A \in \mathcal{A}$, then $f^{-1}(A) \in \mathcal{F}$ for every $A \in...
  30. D

    Can someone help me understand the difference between a measure, and a function which is measurable?

    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...
  31. M

    Proving Measurability of ##A## from ##E=A \cup B## with ##|B|=0##

    Homework Statement Let ##E \subset \mathbb R^n## be a measurable set such that ##E=A \cup B## with ##|B|=0## (##B## is a null set). Show that ##A## is measurable. The Attempt at a Solution I know that given ##\epsilon##, there exists a ##\sigma##-elementary set ##H## such that ##E \subset...
  32. R

    Distance at which electric field causes measurable change

    hello, I was wondering if there is a way in which it would be possible to calculate the distance at which an electric field would need to be to polarize a neutral object or mass m, to a point where the object being like a rod, aligns with the field. I guess this is dependent on the mass...
  33. G

    Does potential energy have measurable corresponding mass?

    When a rubber band is stretched, or a battery is charged, or two massive objects are separated, the potential energy of all these systems increases in each situation. Now say that any of these systems were suspended in space. If we were to measure the gravitational field of the uncharged...
  34. nomadreid

    Elementary point about measurable cards

    Just refreshing my understanding of measurable cardinals, the first step (more questions may follow, but one step at a time) is to make sure I understand the conditions: one of them is For a (an uncountable) measurable cardinal κ, there exists a non-trivial, 0-1-valued measure μ on P(κ)...
  35. E

    Measurable quantites vs Base units

    So here's the familiar SI base units from NIST length mass time electric current thermodynamic temperature amount (mole) luminous intensity Something has been bugging me about this. For whatever reason I am thinking all quantities are calculated by just three on the list -...
  36. nomadreid

    Questions about real-valued measurable cardinals and the continuum

    Putting the following three statements together: (a) Assuming that the continuum hypothesis is false, the power of the continuum 2\aleph0 is real-valued measurable. (b) The existence of a real-valued measurable and the existence of a measurable (= real-valued measurable & inaccessible)...
  37. J

    How measurable function spreads intervals

    Assumptions: f:[a,b]\to\mathbb{R} is some measurable function, and M is some constant. We assume that the function has the following property: [x,x']\subset [a,b]\quad\implies\quad |f(x')-f(x)|\leq M(x'-x) The claim: The function also has the property m^*(f([a,b]))\leq M(b-a) I'm not...
  38. Fredrik

    Closures of the set of measurable functions

    Can a measurable function be a.e. equal to a non-measurable function? Let ##(X,\Sigma,\mu)## be an arbitrary measure space. Let M be the set of measurable functions from X into ##\mathbb C##. I know that M is closed under pointwise limits. I'd like to know if M is also closed under the types...
  39. F

    Measurable and integrable

    Homework Statement Let f : (0,1) —>R be measurable( w.r.t. Lebesgue measure) function in L1((0,1)). Define the function g on (0,1)× (0,1) by g(x,y)=f(x)/x if 0<y<x<1 g(x,y)=0 if 0<x≤y<1 Prove: 1) g is measurable function (w.r.t. Lebesgue measure in the prodcut (0,1)× (0,1) 2)g is integrable...
  40. F

    Does If f be a Measurable Function Imply Finite ∫|f|dm?

    If f be a measurable function. Assume that lim λm({x|f(x)>λ}) exists and is finite as λ tends to infinite Does this imply that ∫|f|dm is finite? Here m is the Lebesgue measure in R If not can anyone give me an example??
  41. S

    Exploring the Smallest Measurable Units: Planck Length & Pi

    Hi all. This is my first time posting so forgive me If I am doing something wrong. I am a year 7 student interested in all types of physics and my question is, if nothing can be smaller than Planck length then wouldn't past a certain point the digits of pi become obsolete? Simply because the...
  42. H

    Measurable with Respect to Complete Space

    Homework Statement Let f:(X,A,μ)->[0,infinity] have a Lebesgue integral, meaning that the inf(upper lebesgue sum)=sup(lower lebesgue sum)=L for a finite L. Show that f is measurable with respect to the completion of the sigma algebra A with respect to μ. You may fix an integrable set E...
  43. R

    MHB Approximation property with F sigma and G delta Sets to show a set is measurable

    Prove that a set $A\subset\mathbb{R}^n$ is (Lebesgue) measurable $\iff$ there exist a set $B$ which is an $F_{\sigma}$ and a set $C$ which is a $G_{\delta}$ such that $B\subset A\subset C$ and $C$~$B$ (C without B) is a null set. $F_{\sigma}$ is a countable union of closed sets, and...
  44. S

    Proving Measurable Functions Convergence in Finite Measure Sets

    Let E be of finite measure and let \{ f_{n} \} _{n \geq 1} : E \rightarrow \overline{\mathbb{R}} measurable functions, finites almost everywhere in E such that f_{n} \rightarrow_{n \to \infty} f almost everywhere in E. Prove that exists a sequence (E_{i})_{i \geq 1} of measurable sets of E such...
  45. F

    Is A a Measurable Set with Sandwich Property?

    Suppose that A is subset of R (real line) with the property for every ε > 0 there are measurable sets B and C s.t. B⊂A⊂C and m(C\B)<ε Prove A is measurable By definition A is measurable we need to prove m(E)=m(E∩A)+m(E\A) for all E the ≤ is trivial enough to show ≥: Since C is...
  46. O

    MHB Proving that the sum of 2 measurable functions is measurable

    I know there are many proofs for this but I am having trouble proving this fact using my book's definition. My book defines first a non negative measurable function f as a function that can be written as the limit of a non decreasing sequence of non-negative simple functions. Then my book...
  47. S

    Semi continuity and Borel Sets - Measurable Functions

    In a book I'm reading it says: \newline If f: \mathbb{R} \longrightarrow \mathbb{R} is lower semi continous, then \{f > a \} is an open set therefore a borel set. Then all lower semi continuous functions are borel functions. It's stated as an obvious thing but I couldn't prove it. The definition...
  48. A

    Integral Inequality for Measurable Functions

    For what class of functions we have: $$ \int_{\Omega} [f(x)]^m dx \leq C\Bigr ( \int_{\Omega} f(x)dx\Bigr)^{m}, $$ where ##\Omega## is open bounded and ##f## is measurable on ##\Omega## and ##C,m>0##.
  49. C

    Sequence of measurable subsets of [0,1] (Lebesgue measure, Measurable)

    Homework Statement Let \left\{E_{k}\right\}_{k\in N} be a sequence of measurable subsets of [0,1] satisfying m\left(E_{k}\right)=1. Then m\left(\bigcap^{\infty}_{k=1}E_{k}\right)=1. Homework Equations m denotes the Lebesgue measure. "Measurable" is short for Lebesgue-measurable. The Attempt...
  50. TheBigBadBen

    MHB Measurable Function (Another Question)

    Is it true that if f:\mathbb{R}\rightarrow\mathbb{R} is a measurable function and E\subset\mathbb{R} is measurable, then f(E) is measurable? What if f is assumed to be continuous? I think that the answer is no for the first and yes for the second, but I have no idea how to prove/disprove either.