• a \}[/itex] is measurable in the sense of Lebesgue. (physicsforums.com)
  • For example: \emph{a function $f:[a,b] \to {\mathbb R}$ is ${\mathcal S}_oACG$ on a Lebesgue measurable set $E \subset [a,b]$ $\Longleftrightarrow$ ${\mathcal S}_o\text{-}\mu_f$ is absolutely continuous on $E$ $\Longleftrightarrow$ $f$ is $VB^*G \cap (N)$ on $E$ and $f$ is continuous at each point of $E$} (see Theorem 8.4). (projecteuclid.org)
  • In Theorem 5.1, we show that \emph{a function $f:[a,b] \to {\mathbb R}$, Lebesgue measurable on $E \subseteq [a,b]$, is ${\mathcal S}_{ap}ACG$ on $E$ if and only if ${\mathcal S}_{ap}\text{-}\mu_f$ is absolutely continuous on $E$}. This result remains true if ${\mathcal S}_{ap}$ is replaced by ${\mathcal S}_{\alpha,\beta}$ with $\alpha,\beta \in (1/2,1]$ (see Theorem 5.1). (projecteuclid.org)
  • The Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space . (infomutt.com)
  • the volume or measure of the Lebesgue measurable set A is denoted by λ( A ). A Lebesgue measure of ∞ is possible, but even so, not all subsets of R n are Lebesgue measurable. (infomutt.com)
  • If A is Lebesgue measurable, then so is its complement. (infomutt.com)
  • λ( A ) ≥ 0 for every Lebesgue measurable set A . (infomutt.com)
  • If A and B are Lebesgue measurable and A is a subset of B , then λ( A ) ≤ λ( B ). (A consequence of 2, 3 and 4. (infomutt.com)
  • Countable unions and intersections of Lebesgue measurable sets are Lebesgue measurable. (infomutt.com)
  • a in A }, is also Lebesgue measurable and has the same measure as A . (infomutt.com)
  • The Lebesgue measure also has the property of being σ-finite . (infomutt.com)
  • is the Lebesgue measure. (victorchen.org)
  • We will use these facts about the Lebesgue measure throughout. (victorchen.org)
  • The area of a subset of the plane is determined using the Lebesgue measure in analysis, however not all subsets are measurable. (wiexi.com)
  • My try for the question: Let $X,Y$ be subsets of $\mathbb{N}$. Let $f: X\rightarrow Y$ be a bijection function with $\operatorname{Card}(X)=n,\operatorname{Card}(Y)=m$ both in $\mathbb{N}$. I think that If we show Card$(X)$=Card$(Y)$ then we are done. (stackexchange.com)
  • For instance, in 2, if $S=\mathbb R^T$ and we take the map to be measurable with respect to the product $\sigma$-field, then as you pointed out $\{\omega:t\mapsto X_t(\omega)\text{ is continuous}\}$ is not measurable (assuming $T=[0,1]$ or something like that). (stackexchange.com)
  • Let $I\subset \mathbb R$ be an interval. (encyclopediaofmath.org)
  • However, the fundamental theorem of calculus does not apply in this case, i.e. there are continuous functions $f:[a,b]\to\mathbb R$ of bounded variation such that the identity \[ f(b') - f(a') =\int_{a'}^{b'} f' (t)\, dt \] fails for a set of pairs $(b', a')\in I\times I$ of positive measure (see below in the section Examples ). (encyclopediaofmath.org)
  • In reality, I am interested in Borel sets on a locally compact space $X$. I can also assume that the support of the measure is $X$, if it helps. (mathoverflow.net)
  • A $\sigma$-regular Borel measure on a $\sigma$-finite $X$ is 'regular' (Lang, p. 257), so the measure of any Borel set is sup of measures of compacts inside it. (mathoverflow.net)
  • The case $m=0$ gives atomic measurable spaces (disjoint unions of points), whereas $m=ℵ_0$ gives disjoint unions of real lines (alias standard Borel spaces). (mathoverflow.net)
  • A nonincreasing function is Borel measurable. (metamath.org)
  • A classical result of Norbert Wiener characterises doubly shift-invariant subspaces for square integrable functions on the unit circle with respect to a finite positive Borel measure μ, as being the ranges of the multiplication maps corresponding to the characteristic functions of μ-measurable subsets of the unit circle. (lse.ac.uk)
  • It is shown that if A and B are Borel subsets of ℝ 2 n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these planes have positive Hausdorff m -measure. (degruyter.com)
  • X:\Omega\rightarrow \mathcal{Y} $$ Now, we have that the law of $X$ is a Borel probability measure, which has (perhaps) nicer properties than in the $\Bbb{R}^T$ picture. (stackexchange.com)
  • A localizable measurable space is a measurable space so equipped, such that every collection of measurable sets has an essential union. (ncatlab.org)
  • be a collection of measurable sets. (victorchen.org)
  • Thus isomorphism classes of measurable spaces are in bijection with functions M: Card'→Card, where Card denotes the class of cardinals and Card' denotes the subclass of Card consisting of infinite cardinals and 0. (mathoverflow.net)
  • M[/itex], then this set has to have null measure, because either the functions doesn't converge here or they're not finite. (physicsforums.com)
  • The sum of two sigma-measurable functions is measurable. (metamath.org)
  • Assume that $X$ is a Banach space of measurable functions for which Komlós' Theorem holds. (arxiv.org)
  • Measure and dimension functions: measurability and densities. (degruyter.com)
  • In measure theory, the notion of measurability restricts sets and functions so that limit operations are sensible. (victorchen.org)
  • Every convergent sequence of measurable functions is nearly uniformly convergent. (victorchen.org)
  • X:\Omega\times T\rightarrow\Bbb{R} $$ where $(\Omega,\mathcal{F},\Bbb{P})$ is a probability space and $T$ is a parameter set, typically taken as a subset of $\Bbb{R}^d$ (though, for generalized processes, $T$ is a set of test functions). (stackexchange.com)
  • However, it is usually stated that this probability measure is fairly useless because the sigma algebra on $\Bbb{R}^T$ is somehow not 'rich' enough to ask interesting questions (for instance, the set of continuous functions is not measurable). (stackexchange.com)
  • It is a theorem, I believe, that every localizable measurable space is capable of supporting a semifinite, hence localizable, measure with the same full/null sets. (ncatlab.org)
  • Geometry of sets and measures in Euclidean spaces , volume 44 of Cambridge Studies in Advanced Mathematics . (degruyter.com)
  • with positive measure essentially contains a measurable set with finite positive measure. (ncatlab.org)
  • The underlying measurable space of a paracompact Hausdorff smooth manifold $M$ satisfies the above conditions if and only if $M$ is second countable, i.e., the number of its connected components is finite or countable. (mathoverflow.net)
  • This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ 2 n , as well as dimension of intersections of sets with isotropic planes. (degruyter.com)
  • We associate to any closed convex bounded subset $C$ of $X$ a coefficient $t(C)$ which attains its minimum value when $C$ is closed for the topology of convergence in measure and we prove some fixed point results for affine Lipschitzian mappings, depending on the value of $t(C)\in [1,2]$ and the value of the Lipschitz constants of the iterates. (arxiv.org)
  • Every measurable space canonically splits as a disjoint union of its ergodic subspaces, i.e., measurable spaces that do not admit measures invariant under all automorphisms. (mathoverflow.net)
  • The measurable space represented by $(m,n)$ is the disjoint union of $n$ copies of $2^m$ , where $2=\{0,1\}$ is a measurable space consisting of two atoms and $2^m$ denotes the product of $m$ copies of 2. (mathoverflow.net)
  • a measurable set can be approximated by a disjoint collection of cubes. (victorchen.org)
  • In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set - the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ∞ {\displaystyle \infty } if the subset is infinite. (wikipedia.org)
  • The counting measure can be defined on any measurable space (that is, any set X {\displaystyle X} along with a sigma-algebra) but is mostly used on countable sets. (wikipedia.org)
  • that is, all subsets of X {\displaystyle X} are measurable sets. (wikipedia.org)
  • displaystyle \vert A\vert } denotes the cardinality of the set A . {\displaystyle A.} The counting measure on ( X , Σ ) {\displaystyle (X,\Sigma )} is σ-finite if and only if the space X {\displaystyle X} is countable. (wikipedia.org)
  • Taking f ( x ) = 1 {\displaystyle f(x)=1} for all x ∈ X {\displaystyle x\in X} gives the counting measure. (wikipedia.org)
  • Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (metamath.org)
  • 𝐋^p(X)$ is separable if and only if every (semifinite) measure on $X$ is $σ$ -finite. (mathoverflow.net)
  • There is a complete classification of measurable spaces up to isomorphism. (mathoverflow.net)
  • of localizable measurable spaces. (ncatlab.org)
  • Healthy People 2010 contains 467 objectives that identify specific measures to monitor health in the first decade of the 21st Century. (cdc.gov)
  • Objectives may include more than one measure. (cdc.gov)
  • Some objectives include several measures, or sub-objectives. (cdc.gov)
  • National data are available for all of the measurable objectives. (cdc.gov)
  • State data are available for a subset of the measurable objectives. (cdc.gov)
  • Age adjustment groups for Healthy People age-adjusted measures and ICD 10 codes for cause-specific Healthy People mortality objectives are delineated with great detail in DATA2010. (cdc.gov)
  • These objectives should be specific, measurable, and time-bound. (coactiveevents.com)
  • Quantitative research - With more-structured methods such as surveys, you gather measurable data about what users do and test assumptions you developed from qualitative research. (interaction-design.org)
  • Other targets, such as killing 2 million feral cats by 2020, were not explicitly linked to measurable conservation outcomes, such as an increase in populations of threatened native animals. (edu.au)
  • Pip (counting) - Easily countable items Set function - Function from sets to numbers Counting Measure at PlanetMath. (wikipedia.org)
  • This is a preorder on the measurable sets. (ncatlab.org)
  • is localizable depends only on which measurable sets are full (or null ). (ncatlab.org)
  • ideal of null sets), without equipping it with the measure of any other set. (ncatlab.org)
  • On the fundamental geometrical properties of linearly measurable plane sets of points. (degruyter.com)
  • The "strange" behavior of non-measurable sets gives rise to such statements as the Banach-Tarski paradox , a consequence of the axiom of choice . (infomutt.com)
  • EmilioPisanty I have to compute the Jordan measure of Cantor sets. (stackexchange.com)
  • The probability measure $\Bbb{P}$ induces a probability measure $\Bbb{P}_X$ on the cylinder sets of $\Bbb{R}^T$ through the pushfoward. (stackexchange.com)
  • A function is measurable iff the preimages of all left-closed intervals unbounded above are in the subspace sigma-algebra induced by its domain. (metamath.org)
  • We include data from our longitudinal population-based study of diabetes and its complications, studies on the role of insulin resistance and insulin secretion in the pathophysiology of type 2 diabetes, and in-depth studies of diabetic kidney disease that include measures of glomerular function and research kidney biopsies. (bvsalud.org)
  • Each of the Core Web Vitals represents a distinct facet of the user experience, is measurable in the field , and reflects the real-world experience of a critical user-centric outcome. (wpengine.com)
  • The 39 tracer indicators measure coverage of a subset of services that reflects various dimensions of the overall coverage of health services. (who.int)
  • Beyond defining biological coverage in a measurable way, the proportion of blood meals obtained from humans and the proportion of bites upon unprotected humans occurring indoors also suggest optimal target product profiles for delivering insecticides to humans or livestock. (biomedcentral.com)
  • The combined measure is defined as the proportion of a population with service coverage and not experiencing financial hardship due to large spending on health in a country in a given year. (who.int)
  • By definition, every measurable set is already arbitrary close in "volume" to an open set. (victorchen.org)
  • and 3) measuring total and allergen-specific immunoglobulin E (IgE) from a blood sample drawn at the NHANES mobile examination site). (cdc.gov)
  • Let $X$ be a measure space. (mathoverflow.net)
  • The underlying measurable space of a locally compact group $G$ satisfies the above conditions if and only if $G$ is second countable as a topological space. (mathoverflow.net)
  • Of course, a set equipped with a localizable measure is a localizable measure space . (ncatlab.org)
  • hence an arbitrary von Neumann algebra may be viewed as a noncommutative localizable measurable space (in the sense of noncommutative geometry ). (ncatlab.org)
  • The ( φ , k ) rectifiable subsets of n -space. (degruyter.com)
  • Typically, researchers begin with qualitative measures, to discover users ' needs and motivations . (interaction-design.org)
  • To suppress mosquito populations, vector control products targeting their blood hosts must attain high biological coverage of all available sources, rather than merely high demographic coverage of a targeted resource subset, such as humans while asleep indoors. (biomedcentral.com)
  • Our data revealed that resveratrol induced measurable changes in circulating immune cells in humans and showed considerable antioxidant activities and other biological effects in humans in vivo. (hindawi.com)
  • Users can also obtain data for the measures used to track the 10 Healthy People 2010 Leading Health Indicators. (cdc.gov)
  • The researchers used kelp records from a 34-year time series of data taken by Landsat satellites, which - among many other characteristics - measured kelp canopies. (joannenova.com.au)
  • While the median (range) duration of virus excretion in stools and sputa for the entire measurable cohort (n = 56) was 27 (16-127) and 21 (14-52) days, respectively ( 3 ), it was 125 and 16 days for patient 1, and 109 and 52 days for patient 3 (viral excretion data could not be obtained from patient 2 because sequential specimens for detection were unavailable). (cdc.gov)
  • However, it is unknown whether resveratrol exerts measurable immunomodulatory effects in humans. (hindawi.com)
  • Our results demonstrate sex differences in some oxycodone self-administration measures. (frontiersin.org)
  • As such, it has the potential to transform the way WHO works by anchoring its commitments in measurable results, and thereby increasing the likelihood that the GPW 13 triple billion targets will be met. (who.int)
  • They might later test their results by using quantitative measures . (interaction-design.org)
  • The service coverage measure will combine the component tracer indicators into an overall index. (who.int)
  • The IgE studies, as a part of Allergy Component, measured both total IgE and also allergen-specific IgE responses to the same panel of allergens measured in the household dust collection. (cdc.gov)
  • A nationally representative sample of household dust was collected to measure the amounts of common indoor allergens present which may be related to asthma and other allergic disease. (cdc.gov)
  • Selecting 'Objective Operational definitions by Focus Area' will provide a list of operational definitions for each Healthy People 2010 measure. (cdc.gov)
  • Existing Spreadsheets and Technical Information Selecting 'Objective Operational definitions by Focus Area' will provide a list of operational definitions for each Healthy People 2010 measure. (cdc.gov)
  • Although the majority of behavioral measures associated with oxycodone self-administration did not show sex differences on their own, when analyzed together using partial least squares regression, their relative contributions to reinstatement were sex-dependent. (frontiersin.org)
  • Without loss of generality we can assume that the support of the measure equals $X$ (i.e., the measure is faithful), because we can always pass to the subspace defined by the support of the measure. (mathoverflow.net)
  • Each objective includes a statement of intent, a baseline value for the measure to be tracked, and a target to be achieved by the year 2010. (cdc.gov)
  • The indicator is a comparative measure to establish the GPW 13 baseline reporting and will facilitate cross-country comparisons and comparisons within countries over time. (who.int)
  • A combined measure of service coverage and related financial hardship will be used to monitor progress towards the GPW 13 targets. (who.int)
  • a) The healthy life expectancy (HALE) indicator, an overarching integrative indicator and a summary measure of average levels of population health, will be used to report the overall progress towards the combined GPW 13 triple billion targets. (who.int)
  • Each of the triple billion targets will be measured using composite indices. (who.int)
  • Core Web Vitals are the subset of Web Vitals that apply to all web pages, should be measured by all site owners, and will be surfaced across all Google tools. (wpengine.com)
  • Measure Theory (Fourth ed. (wikipedia.org)
  • Many of the important theorems of measure theory fail to hold in full generality. (ncatlab.org)
  • With these challenges, economic impact assessments of wars often reflect only a subset of the broad and persistent misery engendered by large-scale human violence. (nakedcapitalism.com)
  • The GPW 13 WHO Impact Framework aims to implement the measurable impact for people at the country level. (who.int)
  • The impact of the progress made towards each billion target in contributing to healthy life expectancy will be quantified and standard guidance and technical support will be provided to Member States so they can measure and report on healthy life expectancy themselves. (who.int)
  • And more than four fifths said they had achieved a measurable increase in reliability. (information-age.com)
  • T-lymphocyte subsets were measured 6 months after disease onset by flow cytometry using fluorescein isothiocyanate-labeled specific monoclonal antibodies. (cdc.gov)
  • 2. Participant has measurable disease per RECIST 1.1/iRECIST. (who.int)
  • 3. Participant has at least one tumour lesion amenable to serial biopsies and is willing to provide consent for biopsies and has measurable disease per RECIST 1.1/iRECIST, excluding the lesion(s) identified for biopsy. (who.int)
  • It's easier to measure success when you have a specific goal to aim for. (coactiveevents.com)
  • In the 2005-2006 survey, a basic subset of allergen-specific IgE antibodies were measured for all participants ages 1+ year (Group 1). (cdc.gov)
  • Caring therefore is not a subset of medicine, instead medicine is one of the tools used for caring. (dcscience.net)