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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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 }...
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...
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...
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...
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?
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...
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...
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?
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...
< 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...
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...
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...
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...
**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...
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...
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...
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...
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...
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)...
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 -...
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)...
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...
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...
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...
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??
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...
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...
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...
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...
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...
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...
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...
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##.
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...
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.