Measure Theory - Science topic

Let n ∈ N where set A ⊆ Rⁿ. Suppose a set A is unbounded, if for any r>0 and x₀ ∈ Rⁿ, d(x,x₀)>r for some x ∈ A, where d is the standard Euclidean Metric of Rⁿ.

If U is the set of all unbounded A measurable in the Caratheodory sense, using the Hausodorff Outer measure (pg. 2, def. 2 of paper), and the mean of A is taken w.r.t the Hausdorff measure and dimension (pg. 2-3, def. 3-4), then prove:

The mean of A is finite, for a subset of U with a cardinality only less than |U|.

(See attached article for more info).

Let (X,d) be a metric space. If set A⊆X, let H^α be the α-dimensional Hausdorff measure on A, where α∈[0,+∞) and dim_H(A) is the Hausdorff measure of set A.

In addition, when n∈N, with set A⊆Rⁿ and function f:A→R; consider the following definitions:

If we define a sequence of sets where (F_r*)_r∈N, where if h is the dimension function (https://en.wikipedia.org/wiki/Dimension_function), then when:

then the generalized expected value of f w.r.t (F_r*)_r∈N is E^**[f,F_r*] where:

∀(ε>0)∃(N₁∈N)∀(r∈N)(r≥N₁ ⇒ 1/H^h(dom(F_r*)) ∫_dom(Fr*) f_r* dH^h - E^**[f,F_r*] < ε)

such that for set V, we want the generalized expected value to exist for all f∈V w.r.t at least one sequence (in a set of sequences of sets) where

Motivation: I want a choice function which chooses a unique set (of equivelant sequences of sets) where the generalized expected value of f w.r.t each sequence has a finite value for all f in a prevelant (https://en.wikipedia.org/wiki/Prevalent_and_shy_sets) subset of R^A. Note, however, the choice should be natural, i.e. consider the question below:

Question: Does there exist a choice function that chooses a unique set (of equivalent sequences of sets) where we get each sequence of sets converges to {(x,f(x)):x∈A}, such that:

Notes On Question: If the solution is extraneous, what other criteria should be included to get a unique choice function? (Note if the solution is always extraneous, we want to replace “equivelant sequences of sets” with the following: ”the set of all sequences of sets, where the generalized expected values of f w.r.t each sequence is the same”.)

If the answer is too complicated to post, answer here (https://mathoverflow.net/q/449272/504799). Note with the link, dimension function (https://en.wikipedia.org/wiki/Dimension_function) h shouldn't only be of Fᵣ*; otherwise, the generalized expected value would be undefined (e.g. when H^h(dom(F_r*)), h is the dimension function of dom(F_r*), not F_r*).

As a researcher, often, it comes to my mind why do we require measure theory and what is the significance and applications of measure theory, measurable set, space and function in real life. Can anyone help me and enlighten me in this direction.

When a set of functions with a property is a prevalent subset of some other set of functions, almost all functions with that property belong to that set of functions.

Is it true the set of all measurable functions (using the uniform probability measure for sets measurable in caratheodory sense) with infinite or undefined expected values form a prevalent subset of the set of all measurable functions?

This is in context of the objective function of a multivariate optimization problem say, f(a,b,c).

I am looking for a "measure" for the degree of bias of f(a,b,c) towards any of the input variables.

Hi, RG community! I am new to network analysis and I am currently facing a challenge with coding, processing, and quantifying networks in a hierarchical scheme. In this scheme, nodes pertain to differing hierarchical ranks and ranks denote inclusion relationships. So, for example if node “A” includes node “Z”, it is said that “A” is “Z”’s parent and “Z” is “A”’ daughter. However, a rather uncommon feature is that nodes at different ranks of the hierarchy can relate in a non-inclusive fashion. For example, node “A” parent of “Z” may have a directional link to “Y”, which is “B”’s daughter (if “A” were directionally linked to “B”, then it could be said that “A” is “Y”’ aunt). Here is a more concrete example to illustrate the plausibility of this scheme: “A” is a website in which person “Z” is signed in (inclusiveness; specifically, parentship); website “A” can advertise banners of website “B” (siblingship) or recommend to follow a link to person “Y” profile in website “B” (auntship).

OK. So, in the image below (left top panel) I present a graphical depiction of this rationale. For simplicity, a two-rank hierarchy is used, where gray and red colors denote higher and lower hierarchies, respectively. The image displays siblingship, parentship, and auntship links. My first approach to coding this network scheme was to denote inclusiveness as one-directional relationships (green numbers) and simple links as symmetrical (two-way; brown numbers) relationships (see table in right panel). However, I soon realized that this does not reflect what I expected in networks’ metrics. For example, I am mainly interested in quantifying cohesiveness and the way I coded the network in left top panel entails something like the non-hierarchical network depicted at the left bottom panel. In short, I am not interested in the directionality of the links but in actual inclusiveness. To my mind, the network in the top panel is more cohesive than that in the bottom panel but my coding approach does not allow me to distinguish between them formally.

The solution conceived in the interest of solving this problem was to stipulate that a relationship between any pair of nodes implicates a relationship of each with all of the other’s descendance. This certainly yields, for example, the top network being more cohesive than that in the bottom, which is in line with my goals. However, this solution is not at all as elegant as I would have hoped. Can anyone tell if there is a better solution? Maybe another way to code or an R package allowing for qualitatively distinct relationships (not just one-way or two-way). Thank you.

I have yet to understand amenability and group theory, but the questions in my attachment might be of interest.

I would be glad if all the questions were answered; however, if you wish for a particular question, read section 4 of my paper and the question in section 5.

I want an elegant choice function where, for specific A in 4.4, gives the structure (defined in 4.2) that I'm looking for.

APA says, “validity refers to the degree to which evidence and theory support the interpretation of test scores for proposed uses of tests”. However, questionnaires with independent items (some items about intensity, others about frequency, others nominals), that do not generate a score, also need validation (at least a content validation). So can I ensure that validation is just possible to support the interpterion of test scores?

In the 21st century, we can safely say that absolutely all modern achievements in the field of science are based on the successes of modeling theory, on the basis of which practical recommendations are given that are useful in physics, technology, biology, sociology, etc., are extracted. In addition, this is due to the fact that the application of the principles of the theory of measurements in determining the fundamental constants allows us to check the consistency and correctness of the basic physical theories. In addition to the above, the quantitative predictions of the main physical theories depend on the numerical values ​​of the constants included in these theories: each new sign can lead to the detection of a previously unknown inconsistency or, conversely, can eliminate the existing inconsistency in our description of the physical world. At the same time, scientists have come to a clear understanding of the limitations of our efforts to achieve very high measurement accuracy.

The very act of measurement already presupposes the presence of a physical and mathematical model that describes the phenomenon under study. Simulation theory focuses on the process of measuring the experimental determination of values ​​using special equipment called measuring instruments. This theory covers only aspects of data analysis and the procedure for measuring the observed quantity or after the formulation of a mathematical model. Thus, the problem of uncertainty before experimental or computer simulations caused by the limited number of quantities recorded in the mathematical model is usually ignored in measurement theory.

Is the reciprocal of the inverse tangent $\frac{1}{\arctan x}$ a (logarithmically) completely monotonic function on the right-half line?

If $\frac{1}{\arctan x}$ is a (logarithmically) completely monotonic function on $(0,\infty)$, can one give an explicit expression of the measure $\mu(t)$ in the integral representation in the Bernstein--Widder theorem for $f(x)=\frac{1}{\arctan x}$?

These questions have been stated in details at the website https://math.stackexchange.com/questions/4247090

Psychometrics is a field of study concerned with the theory and technique of psychological measurement. As defined by the US National Council on Measurement in Education (NCME), psychometrics refers to psychological measurement. Generally, it refers to the specialist fields within psychology and education devoted to testing, measurement, assessment, and related activities.

The field is concerned with the objective measurement of skills and knowledge, abilities, attitudes, personality traits, clinical constructs and mental disorders as well as educational achievement. Some psychometric researchers focus on the construction and validation of assessment instruments such as questionnaires, tests, raters' judgments, psychological symptom scales, and personality tests. Others focus on research relating to measurement theory (e.g., item response theory; intraclass correlation).

Practitioners are described as psychometricians. Psychometricians usually possess a specific qualification, and most are psychologists with advanced graduate training in test interpretation, psychometrics, and measurement theory. In addition to traditional academic institutions, many psychometricians work for the government or in human resources departments. Others specialize as learning and development professionals.

Thank you for your comment on my question.

The topic of the influence of the leader occupies a major part in management and other literature such as sociology, politics, etc.,

Am I looking for a "scale"? Or a "questionnaire," any other method, to measure the leader-triggered effect on a team. ?

Thanks for your time and contribution.

How to prove or where to find the integral inequality (3.3) involving the Laplace transforms, as showed in the pictures here? Is the integral inequality (3.3) valid for some general function $h(t)$ which is increasing and non-negative on the right semi-axis? Is the integral inequality (3.3) a special case of some general inequality? I have proved that the special function $h(t)$ has some properties in Lemma 2.2, but I haven't prove the integral inequality (3.3) yet. Wish you help me prove (3.3) for the special function $h(t)$ in Lemma 2.2 in the pictures.

The development of IT and information technologies increasingly affects economic processes taking place in various branches and sectors of contemporary developed and developing economies.

Information technology and advanced information processing are increasingly affecting people's lives and business ventures.

The current technological revolution, known as Industry 4.0, is determined by the development of the following technologies of advanced information processing: Big Data database technologies, cloud computing, machine learning, Internet of Things, artificial intelligence, Business Intelligence and other advanced data mining technologies.

In connection with the above, I would like to ask you:

How to measure the value added in the national economy resulting from the development of information and IT technologies?

Please reply

Best wishes

suppose in 2 dimensional space there are two triangles. Now there are different cases as follows

1> they have no intersection

2> they intersect on a point

3> they intersect on an edge

4> they intersect on a 2D area

I want to quantify these intersections using a measure defined on sets. Is there a measure that can be helpful in this pursuit.

Regard and thanks in advance!!!

Zubair

It is known that the FPE gives the time evolution of the probability density function of the stochastic differential equation.

I could not see any reference that relates the PDF obtain by the FPE with trajectories of the SDE.

for instance, consider the solution of corresponding FPE of an SDE converges to pdf=\delta{x0} asymptotically in time.

does it mean that all the trajectories of the SDE will converge to x0 asymptotically in time?

A new paper in "Constructive Mathematical Analysis" by Prof. Michele Campiti.

You can download the paper for free:

https://dergipark.org.tr/cma/issue/47453/574194

Recent research has shown key important aspects with measurability of theoretical postulates verifying hypotheses parameters, processes, phenomena, and models.

Tensor matrices have necessarily roles to bring about complex nature manifesting spatially and temporarily. Explaining everything that is existing in terms of the fundamental entities have lead to realization of geometric topology space tensor manifold time evolving event gridnetwork.

Einstein's General Theory of Relativity, Quantum to Particle Theory of Everything, String Theory among others have measurability in mind a proof of model requirement automatically. For example, Schwartzchild blackhole mathematics helped to identify, observe, and measure singularity blackhole consequently, the recent telescopic photos observing directly, proving validity with General Theory of Relativity tensor predictive capability.

Providing the thumb rules, below certain associative relationships might connect mathematics with physics to measure model......

. typically scalars, scalar matrices are helpful to get statistical measurements that are analyzable observationally experimentally......

. tensors have stochastical vector matrices that aren't amenable to direct measurements. Hence transforming tensors or matrix tensors to scalar matrix systems are key to make measurable operational parametric graphical experimental observational gridnetworks.

My analyses metrix protocol techniques have yielded a rough estimate of overall globally ~80% of objects universally are measurable statistically. This will mean ~20% are uncertainity stochastic probabilities with a few% inherent immeasurable tensor network, aether may be example. I have space time sense 2x2 tensor grid part of a large tensor matrix that if transformed to 5 dimensional like scalar matrix natural manifolds protocol will help eventually in the quantitative grand unified theory of everything. There are more to come after our QFM/EM modeling going on with collaborative platform TEI.

Given below are a few references that are associated, not exhaustive, suggestions welcome. Additions editions expansions!!!!!

(1) https://www.researchgate.net/project/TOWARDS-THE-GRAND-UNIFIED-THEORY-OF-EVERYTHING-LINKING-FUNCTIONALITY-MODEL-CONCRETE-TO-ABSTRACTIONS?_sg=XNNU-DQ2rx6YNmUfxUjYZ6LaLamHZK8jGl_UOQblslDC1A9ygt_x0mY-iylt49cJFCqpUNKip5okOdqjIf43mH-lRm2JzYvjuM4j including working papers with tensor analyses......

(2)

(3)

(4)

(5) , , , & https://www.researchgate.net/project/A-Synthetic-Magnetic-Unipolar-Array-Not-a-Dirac-Monopole?_sg=zOOB5jYpcBuzrnEjYVwI8d7oSR2-UNEsI1Yu3Jf2e2uQQ9DbTVGxz3rQLfVEU7-n8Y6Fo0ZY-NFy8-gDCORKogPQF099GP5bghsc

(6) Zurek, Wojciech H. (2003). "Decoherence, einselection, and the quantum origins of the classical". Reviews of Modern Physics. 75 (3): 715. arXiv:quant-ph/0105127. Bibcode:2003RvMP...75..715Z. doi:10.1103/revmodphys.75.715 & Dan Stahlke. "Quantum Decoherence and the Measurement Problem" (PDF). Retrieved 2011-07-23.

(7)

(8) https://www.researchgate.net/post/Measurable_function_Real_analysis?_iepl%5BgeneralViewId%5D=UrJPT0Tf7xZ5zvi3IaLKUrm7s2pwF1roCYYk&_iepl%5Bcontexts%5D%5B0%5D=searchReact&_iepl%5BviewId%5D=DuVPrSZERB3gc1XS1sy4M7xRN2eix4sWRMm6&_iepl%5BsearchType%5D=question&_iepl%5Bdata%5D%5BcountMoreThan20%5D=1&_iepl%5Bdata%5D%5BinteractedWithPosition1%5D=1&_iepl%5Bposition%5D=1&_iepl%5BrgKey%5D=PT%3A5cadcf48aa1f099c5752a2f4&_iepl%5BtargetEntityId%5D=Q%3A5cadcf48aa1f099c5752a2f4&_iepl%5BinteractionType%5D=questionView

I have meet a problem in reading paper " P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana, 1(1):145–201, 1985." My problem is in Lemma I.2, I do not understand clearly the proof of equality v=\sum_{i\in J} v_i\delta_{x_i}. This proof comes from the result as follows:

Let ν be a non-negative bounded measure. Assume that there exists δ > 0 such that for all A Borelian, ν(A) = 0 or ν(A) ≥ δ. Then, there exist {x_i}, i\in J, and ν_i > 0 such that ν = \sum_{i\in J} ν_iδ_{x_i}.

But I can not prove it. I hope that some expert will help me to solve it.

I am looking forward the answer from anyone.

The non measurable set is formed by selecting one element from each equivalence class obtained by the relation x ~ y if x-y is rational.

but we suggest we do not accept this form of axiom of choice applied to collection of sets.

To form an arbitrary Cartesian product one needs an indexed family of sets and an indexing set is necessary.

In the above case it seems that the collection is not indexed . No indexing set and explicit indexing map is there.

So we can not form a nonmeasurable set. Thus Banach -Taraski paradox is absent.

we do not work with arbitrary collections but only indexed families and explicit indexing map.

how to avoid the paradox in measure theory

plane where axiom of choice is not needed is also open

Is the fractal measures the same as the fractal dimension?

For V(Vitali Set), m(V)>0 is obvious but even it is not clear that m(V)=1 or m(V)<1.

Does there exists a function $g \in L^1(R)$ such that it satisfy the following property

(1)$ L^1$ norm of g \leq 1

(2) $\hat{g}(s) =1$ for $s\in [-\delta, \delta]$ and

(3) \hat{g} has compact support ?

More precisely 

Denote $R=\mathbb{R}$, $S(R)= Schwarz Class function on R $

I know that $C_{c}^{\infty}(R) \subset S(R)\subset L^1(R)$ and Fourier transform is bijection on S(R).  So one can start with a function with property $(2)$ and $(3)$ above and take $g$ as a inverse Fourier transform of the function. But I don't know that norm of g can be bounded by 1 ?

Dear all,

For any sigma-finite measure $\mu$ on the Borel subsets of a metric space X (separable if needed), we recall that a $\mu$-continuity set is any Borel set B such that $\mu(boundary of B) = 0$

I would like to find a good reference (book or article) where is treated clearly the proof of the result which establishes that the sigma-algebra generated by the $\mu$-continuity sets is equal to the Borel sigma-algebra.

Sincerely,

What is the dual space of bounded continuous functions defined on a metric space (not necessarily compact)? (i would appreciate to have a reference for the result too)

My terminology here is from "Measure Theory" by JL Doob, Springer Graduate Texts in Mathematics, 1994.  Metric and pseudometric spaces are described on pages 3 to 5.

My question is what happens if the triangle inequality is an equality so we have the distance formula d(s, u) = d(s, t) + d(t, u).  This can only be true if every distance is 1, and 1 = 1 + 1 because every a = a².

Now if the pseudometric distance formula is d(s, s) = 0, does this follow directly from the triangle equality or is there something else required?  Is d(s, s) = 0 the same as the Borel function f(x, y) = 0 where x = y?

What I am hoping is that if there is an algebra of subsets where every set is a singleton, then that by itself makes a pseudometric space with a distribution on the Borel Square [0, 1] x [0, 1].  Does the triangle equality automatically make an L-space?

May be it is not that simple. If the triangle equality metric alone does not make a pseudometric space by itself, what does? A field of zeros? What if I have the algebra formed by the union, intersection, and compliment of an elemental set? What if I have a set, a relation, an agumented frame, and a map?

My goal here is to show that music structures are a pseudometric space where a probability distribution is imposed on pitch values by the set construction operation.  This follows my previous question demonstrating that music theory is a 3-fold structure, not two fold as commonly thought.  In a 2-fold structure there is no way to understand why every note does not have a probability of 1/12..

I am conducting a research on how to maximize South-South knowledge exchange in the forest sector between Brazil and Africa, with the case study being Brazil and Mozambique.

It takes a long for results to be seen in the forest sector in any type of initiative, however what are the short and medium term results one could expect in order to know that the knowledge exchange was absorbed and being implemented?

In Alfred Tarski's "Logic, Semantics, Metamathemaitcs" the set of all meaningful sentences is described and the ordinary two-valued system of sentential calculus L is shown to be a subset of all decidable systems that is consistent and complete.  That is amazing!

I would like to compare the algebra of subsets of S that is closed to the set=theoretic operations of union, intersection, and complimentation to the set of sentences of S that is closed to semantic the operations of implication and detachment.

I was thinking that since Tarski shows that the Set of all sentences is a subset of all linear orders, that there is a relation of sentences as linear sequences to the general projective plane of all linear operations.  I think that is the classifying space.

I wonder if I am in correct in assuming that the set of all sentences implies there is a subspace topology in which every element is a singleton with indentically zero.

Is it possible that there is a way to connect the sentences and the algebra of prime ideals by an elemental formula.  Could I used the Serpinski dual operators? The standard model? Maybe the relation is tautologically obvious and every one already knows?

What I realy want is the classifying space for music theory.

If you don't have Tarski's book here an outline of the definition of L with substitution of variables for music:

The ordinary two-valued system L of sentential calculus is the set of all sentences that satisfy the matrix M = [A, B, f, g] where A = {0}, B = {1}, and the function f and g are defined by the implication and detachment formulas: 1) f(0, 0)= f(0,1)=f(1, 1) = 1, f(1, 0) = 0; and 2) g(0) = 1, g(1) = 0. From this definition, it immediately follows that ordinary sentence calculus is a completely defined and axiomatizable system.  The function f is the implication operation used to make new sentences, since {f(0, 0)= f(0,1)=f(1, 1) = 1, f(1, 0) = 0} is the propositional calculus for implication.  The f is the pitch-position equivalence relation given by f(x, y)= 0 where x = y are any two harmonic values defined by the value function, described by Tarski, so that the proposition “if a, then b” is interpreted to signifying the implication “if pitch then position.” The pitch values are frequency values related to positions which are a state of system values in a finite state machine. The pitch value set is defined by integers i indexing points in the frequency domain, while the position value set is defined by integers j that index points that are not defined directly by frequency values (that is positon values have no frequency attached except by the pitch-position equivalence map which is the primitive equivalence relation defining membership in the semantic system). 

The primitive detachment function g in music is the intonation function, which is a function of pitch (that is, frequency) only.  Because intonation means to sound a fundamental mode of vibration at a specific pitch value, the intonation function g is a witness function that quantifies the musical object for all pitch values. Since tuning is by definition a pitch-position equivalence relation between pitch value index numbers and fret value index numbers, it is already clear in Tarski’s definition that pitch-positon function f and the intonation function g in music are inverse homeomorphisms.

For instance, if we have a musical string with 20 fret positions then we have the string f(x, y) = 0 defined by the pairwise disjoint relation of the pitch value set and the fret value set, which run together like railroad tracks that vanish at infinity.  The pitch-position relation is a constant: the same for every string regardless of pitch.  When a second string is added, the interval between the two string is the only determinant of the system formed by the intersection and union of the strings.

Tablature notation of guitar music is an infinite data strip that passes through the guitar and is output (like a player piano) as the tablature notation of the musical sounds.  The numeric sequences in the tablature are a subset of the algebraic closed field that is formed over the guitar by the tuning vector which is the k = 6 point-wise restriction on the continuous function of pitch.

So we have the algebra of subsets of the musical key (using the prime ideals of the tuning space) that do not contain the music key itself; and we have the set of all sentences of the same set that are written in the algebraic language that satisfies the guitar tuning theory (x₁, ... x₆) tuple.

 I know this is correct and sooner of latter, some one will get it.  If you can't answer the question, maybe you can explain why music is not a valid topic in modern mathematics.  The trash that passes for the mathematics of music in the literature is ridiculous.  How come no mathematicians critque these authors like Mazzola "Topos of Music" and Tymoczko "Geometry of Music."

The terminology in this question is taken from "Algebraic Topology" by CRF Maunder. See Problem 9 on Page 59.

"Let H be the abstract 1-dimensional simplicial complex with vertices a⁰, a¹, a², a³, a⁴, a⁵, each pair of vertices being an abstract 1-simplex. Show H has no realization in R²."

Note that I have added a⁵ to the author's text.

The reason I ask is this. We have the guitar tuning as the union of 6 string intervals defined by a 6-tuple representing a point such as (0, 5, 5, 5, 4, 5) or EADGBE, so each string is itself a 1-simplex.  Note that because each tuning interval is already defined on the system fundamental the secondary string spectrum is already inside the system fundamental spectrum.  When the strings are subsets of the system fundamental we say the guitar is "in tune".

If the vertices are the fundamentals of six guitar strings and the each string is defined by an interval between the fundamental state of system and the string, so the interval is always a known as a whole prime number defined on the system fundamental, then does not the solution to the above problem show that guitar music cannot be realized in R²?

I would like to prove in general the structure of music is 3-fold and not 2-fold as Euler thought.

Let (X,d) be a metric space, f: X->X be Borel measurable. Let (K(X),d_H) be a space of compact subsets of X with Hausdorff metric. Is f:K(X)->K(X), where f(A)=\cup_{x\in A}{f(x)} Borel measurable? Is there needed separability of X?

How do I calculate this integral in terms of Bessel functions? 

Assume that M is a compact Riemannian manifold with corresponding volume form and corresponding  measure. m (Correspond to the initial metric).

Assume that A is a closed proper subset of M such that for all smooth diffeomorphism f on M we have m(A)=m(f(A)). does this imply that A has zero measure? 

I was working on 2 papers on statistics when I recalled a study I’d read some time ago: “On ‘Rethinking Rigor in Calculus...,’ or Why We Don't Do Calculus on the Rational Numbers’”. The answer is obviously trivial, and the paper was really in response to another suggesting that we eliminate certain theorems and their proofs from elementary collegiate calculus courses. But I started to wonder (initially just as a thought exercise) whether one could “do calculus” on the rationals and if so could the benefits outweigh the restrictions? Measure theory already allows us to construct countably infinite sample spaces. However, many researchers who regularly use statistics haven’t even taken undergraduate probability courses, let alone courses on or that include rigorous probability. Also, even students like engineers who take several calculus courses frequently don’t really understand the real number line because they’ve never taken a course in real analysis.

The rationals are the only set we learn about early on that have so many of the properties the reals do, and in particular that of infinite density. So, for example, textbook examples of why integration isn’t appropriate for pdfs of countably infinite sets typically use examples like the binomial or Bernoulli distributions, but such examples are clearly discrete. Other objections to defining the rationals to be continuous include:

1) The irrational numbers were discovered over 2,000 years ago and the attempts to make calculus rigorous since have (almost) always taken as desirable the inclusion of numbers like pi or sqrt(2). Yet we know from measure theory that the line between distinct and continuous can be fuzzy and that we can construct abstract probability spaces that handle both countable and uncountable sets.

2) We already have a perfectly good way to deal with countably infinite sets using measure theory (not to mention both discrete calculus and discretized calculus). But the majority of those who regularly use statistics and therefore probability aren’t familiar with measure theory.

The third and most important reason is actually the question I’m asking: nobody has bothered to rigorously define the rationals to be continuous to allow a more limited application of differential and integral calculi because there are so many applications which require the reals and (as noted) we already have superior ways for dealing with any arbitrary set.

Yet most of the reasons we can’t e.g., integrate over the rationals in the interval [0,1] have to do with the intuitive notion that it contains “gaps” where we know irrational numbers exist even though the rationals are infinitely dense. It is, in fact, possible to construct functions that are continuous on the rationals and discontinuous on the reals. Moreover, we frequently use statistical methods that assume continuity even though the outcomes can’t ever be irrational-valued. Further, the Riemann integral is defined in elementary calculus and often elsewhere as an integer-valued and thus a countable set of summed "terms" (i.e., a function that is Riemann integrable over the interval [a,b]  is integrated by a summation from i=1 to infinity of f(x^*_I)Δx, but whatever values the function may take, by definition the terms/partitions are ordered by integer multiples of i). As for the gaps, work since Cantor in particular (e.g., the Cantor set) have demonstrated how the rationals “fill” the entire unit interval such that one can e.g., recursively remove infinitely many thirds from it equal to 1 yet be left with infinitely many remaining numbers. In addition to objections mostly from philosophers that even the reals are continuous, we know the real number line has "gaps" in some sense anyway; how many "gaps" depends on whether or not one thinks that in addition to sqrt(-1) the number line should include hyperreals or other extensions of R1. Finally, in practice (or at least application) we never deal with real numbers anyway (we can only approximate their values).

Another potential use is educational: students who take calculus (including multivariable calculus and differential equations) never gain an appreciable understanding of the reals because they never take courses in which these are constructed. Initial use of derivatives and integrals defined on the rationals and then the reals would at least make clear that there are extremely nuanced, conceptually difficult properties of the reals even if these were never elucidated.

However, I’ve been sick recently and my head has been in a perpetual fog from cold medicines, so the time I have available to answer my own question is temporarily too short. I start thinking about e.g., the relevance of the differences between uncountable and countable sets, compact spaces and topological considerations, or that were we to assume there are no “gaps” where real numbers would be we'd encounter issues with e.g., least upper bounds, but I can't think clearly and I get nowhere: the medication induced fog won't clear. So I am trying to take the lazy, cowardly way out and ask somebody else to do my thinking for me rather than wait until I am not taking cough suppressants and similar meds. 

Is there any proof for the equation

where Hⁿ(F) is the n-dimensional hausdorff measure, C_n is the volume of an n-dimensional ball of diameter 1, and volⁿ(F) is the n-dimensional volume , and F is a Borel subset of Rⁿ.

I want to teach real analysis course this semester. I want to teach more about functional analysis and less about real analysis. I need a brief and new book. Could you please guide me?

It is an old Weis-Adler result that the Boole mapping R∋x→x-1/x∈R is a Lebesgue measure preserving and ergodic. What one can state about the mappings R²∋(x,y)→(x-1/y,y+1/x)∈R² and about R²∋(x,y)→(y-1/x,x+1/y)∈R² ?

The Conjecture  was recently claimed in the note attached.

We are evaluating a new e-learning tool for adaptive learning.

To evaluate the new tool, we measure different variables (e.g. motivation, interest, performance, mood) with online questionnaires.

To measure the cognitive load of some tasks we think to use the three items used by Cierniak et al. (2009). For a more objective measure we hope to find a way to measure cognitive load with log file analysis.

Does somebody knew a method do measure cognitive load with log files?

All the books I read get product measures by using results from integration theory. I think that a more direct route should be possible. However countable additivity is not simple to get directly. Are you aware of a reference following a direct route? Thank you in advance

I imagine the following scenario: You have 12 items that measure 2 sub dimensions of a construct, 6 items each construct.

Say the construct is happiness, the sub dimensions were eudaimonic and hedonic happiness. You find the 2 sub dimensions in an exploratory factor analysis, just as theory says.

But the Rasch model does not fit each sub dimension. It fits when you analyze ALL items at once. Can we now combine these insights? E.g. we measured each sub dimension reliably and therefore we can assume that items 1e-6e assess eudaimonia, 1h-6h measure hedonia. These items are then labeled with the respective dimension. In Rasch analysis with all items we would then find that 8.3% (=100%/12) participants were measured well per item. Hedonic item 1h has the largest logit, eudaimonic item 1e the second largest, hedonic 2h the third largest, eudaimonic 2e the fourth etc.

Could we now argue that in the case of happiness the item 1h assesses the most happy persons and this happiness is of hedonic nature?

In case I messed it up or anything is not clear, please ask.

Could we define an integration on a sub-additive measure following the method of what Lebesgue integration was given, i.e using simply the functions?

How do you define an expectation?

Definition of expectation.

We need to have a non-fuzzy mathematical explanation about probability theory and possibility theory. Are they related? If the answer is affirmative, then what is the exact relationship?

Please give your comments on this topic.

I know only the selectivity in analytical spectrometry.

In the paper by Hwa Kil Kim published in June 14, 2012. What is the meaning of (R^D) and the meaning of (J:(R^D)--->(R^D) is a matrix satisfying (

(J_v )_|_v) for all v in (R^D) ). The name of the paper is:Moreau-Yosida approximation and convergence of Hamiltonian systems on Wasserstein space, and it is on RG.

How can we define a measure (or fundamentally an Interval measure) for space with cardinality of higher than R^N (euclidean space) . For example, measure in functional space - are they measurable space at all ?

Generally when a space is measurable the measure is defined by defining Intervals set and defining a measure for that interval. Interval is easily defined for euclidean space but I wonder if the space procedure is used for infinite dimensional space too ?

'The notion of probability does not enter into the definition of a random variable.' (Ref.: page 43 of V. K. Rohatgi and A. K. Saleh, An Introduction to Probability and Statistics, Second Edition, Wiley Series in Probability and Statistics, John Wiley & Sons (Asia) Pte. Ltd., Singapore, 2001.) Here randomness has been defined in the measure theoretic sense.

On the other hand, it has also been said that 'A random variable is a set function whose domain is the elements of a sample space on which a probability function has been defined and whose range is the set of real numbers.' (Ref.: page 9 of J. D. Gibbons and S. Chakraborti, Nonparametric Statistical Inference, Third Edition, Marcel Dekker Inc., New York, 1992.)

One who is not conversant with measure theory would opine that a random variable must be probabilistic. But according to measure theory, a random variable need not be probabilistic, while a probabilistic variable is necessarily random by definition.

There should not be two different definitions of randomness. Perhaps a discussion is needed in this regard.