###### displaystyle

- If m is a Radon measure on X , {\displaystyle X,} then the mapping I : f ↦ ∫ f d m {\displaystyle I:f\mapsto \int f\,dm} is a continuous positive linear map from K ( X ) {\displaystyle {\mathcal {K}}(X)} to R. Positivity means that I(f) ≥ 0 whenever f is a non-negative function. (wikipedia.org)
- Conversely, by the Riesz-Markov-Kakutani representation theorem, each positive linear form on K ( X ) {\displaystyle {\mathcal {K}}(X)} arises as integration with respect to a unique regular Borel measure. (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)
- is the norm on X . {\displaystyle X.} Countably additive vector measures defined on sigma-algebras are more general than finite measures, finite signed measures, and complex measures, which are countably additive functions taking values respectively on the real interval [ 0 , ∞ ) , {\displaystyle [0,\infty ),} the set of real numbers, and the set of complex numbers. (wikipedia.org)
- μ , {\displaystyle \mu ,} viewed as a function from F {\displaystyle {\mathcal {F}}} to the Lp-space L ∞ ( [ 0 , 1 ] ) , {\displaystyle L^{\infty }([0,,} is a vector measure which is not countably-additive. (wikipedia.org)
- Omega )} is finite, the measure μ {\displaystyle \mu } is said to be of bounded variation. (wikipedia.org)
- A set E {\displaystyle E} , not necessarily measurable, is said to be locally measurable if for every measurable set A {\displaystyle A} of finite measure, E ∩ A {\displaystyle E\cap A} is measurable. (wikipedia.org)
- σ {\displaystyle \sigma } -finite measures, and measures arising as the restriction of outer measures, are saturated. (wikipedia.org)
- Countable additivity (or σ-additivity): For all countable collections { E i } i = 1 ∞ {\displaystyle \{E_{i}\}_{i=1}^{\infty }} of pairwise disjoint sets in Σ: μ ( ⋃ k = 1 ∞ E k ) = ∑ k = 1 ∞ μ ( E k ) {\displaystyle \mu \left(\bigcup _{k=1}^{\infty }E_{k}\right)=\sum _{k=1}^{\infty }\mu (E_{k})} One may require that at least one set E has finite measure. (wikipedia.org)
- The motivation for studying empirical measures is that it is often impossible to know the true underlying probability measure P {\displaystyle P} . We collect observations X 1 , X 2 , … , X n {\displaystyle X_{1},X_{2},\dots ,X_{n}} and compute relative frequencies. (wikipedia.org)
- We can estimate P {\displaystyle P} , or a related distribution function F {\displaystyle F} by means of the empirical measure or empirical distribution function, respectively. (wikipedia.org)
- A Borel measure is any measure μ {\displaystyle \mu } defined on the σ-algebra of Borel sets. (wikipedia.org)
- If μ {\displaystyle \mu } is both inner regular and locally finite, it is called a Radon measure. (wikipedia.org)
- The real line R {\displaystyle \mathbb {R} } with its usual topology is a locally compact Hausdorff space, hence we can define a Borel measure on it. (wikipedia.org)
- Also, the Borel measure and the Lebesgue measure coincide on the Borel sets (i.e., λ ( E ) = μ ( E ) {\displaystyle \lambda (E)=\mu (E)} for every Borel measurable set, where μ {\displaystyle \mu } is the Borel measure described above). (wikipedia.org)
- One can define the Laplace transform of a finite Borel measure μ on the real line by the Lebesgue integral ( L μ ) ( s ) = ∫ [ 0 , ∞ ) e − s t d μ ( t ) . {\displaystyle ({\mathcal {L}}\mu )(s)=\int _{[0,\infty )}e^{-st}\,d\mu (t). (wikipedia.org)
- 0. The Cramér-Wold theorem in measure theory states that a Borel probability measure on R k {\displaystyle R^{k}} is uniquely determined by the totality of its one-dimensional projections. (wikipedia.org)
- Let ( X 1 , Σ 1 ) {\displaystyle (X_{1},\Sigma _{1})} and ( X 2 , Σ 2 ) {\displaystyle (X_{2},\Sigma _{2})} be two measurable spaces, that is, Σ 1 {\displaystyle \Sigma _{1}} and Σ 2 {\displaystyle \Sigma _{2}} are sigma algebras on X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} respectively, and let μ 1 {\displaystyle \mu _{1}} and μ 2 {\displaystyle \mu _{2}} be measures on these spaces. (wikipedia.org)
- The uniqueness of product measure is guaranteed only in the case that both ( X 1 , Σ 1 , μ 1 ) {\displaystyle (X_{1},\Sigma _{1},\mu _{1})} and ( X 2 , Σ 2 , μ 2 ) {\displaystyle (X_{2},\Sigma _{2},\mu _{2})} are σ-finite. (wikipedia.org)

###### mathematics

- In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular. (wikipedia.org)
- In mathematics, 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 ∞ if the subset is infinite. (wikipedia.org)
- In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. (wikipedia.org)
- In mathematics, a measure is said to be saturated if every locally measurable set is also measurable. (wikipedia.org)
- In mathematics, a pre-measure is a function that is, in some sense, a precursor to a bona fide measure on a given space. (wikipedia.org)
- In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). (wikipedia.org)
- In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element x or not. (wikipedia.org)
- In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space. (wikipedia.org)

###### finite measure

- It is a generalization of the concept of finite measure, which takes nonnegative real values only. (wikipedia.org)
- δx is a probability measure, and hence a finite measure. (wikipedia.org)
- Suppose that (X, T) is a topological space and that Σ is at least as fine as the Borel σ-algebra σ(T) on X. δx is a strictly positive measure if and only if the topology T is such that x lies within every non-empty open set, e.g. in the case of the trivial topology {∅, X}. Since δx is probability measure, it is also a locally finite measure. (wikipedia.org)

###### Lebesgue

- As a particular case, a measure defined on the Euclidean space Rn is called singular, if it is singular in respect to the Lebesgue measure on this space. (wikipedia.org)
- A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space Rn. (wikipedia.org)
- For instance, the Lebesgue measure of the interval [0, in the real numbers is its length in the everyday sense of the word - specifically, 1. (wikipedia.org)
- 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. (wikipedia.org)
- 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. (wikipedia.org)
- 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. (wikipedia.org)
- 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. (wikipedia.org)
- The Lebesgue σ-algebra is actually the completion of the Borel σ-algebra, which means that it is the smallest σ-algebra which contains all the Borel sets and has a complete measure on it. (wikipedia.org)
- The Lebesgue-Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue-Stieltjes measure, which may be associated to any function of bounded variation on the real line. (wikipedia.org)
- The Lebesgue-Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind. (wikipedia.org)
- If X is n-dimensional Euclidean space ℝn with its usual σ-algebra and n-dimensional Lebesgue measure λn, then δx is a singular measure with respect to λn: simply decompose ℝn as A = ℝn \ {x} and B = {x} and observe that δx(A) = λn(B) = 0. (wikipedia.org)
- More formally, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if its support is at most a countable set. (wikipedia.org)
- Consequently, the completion procedure is needed to extend the Borel measure into the Lebesgue measure, or to extend the product of two Lebesgue measures to give the Lebesgue measure on the product space. (wikipedia.org)
- Take the product X×Y, where X is the unit interval with Lebesgue measure, and Y is the unit interval with counting measure and all sets measurable. (wikipedia.org)
- Then for the minimal product measure the measure of a set is the sum of the measures of its horizontal sections, while for the maximal product measure a set has measure infinity unless it is contained in the union of a countable number of sets of the form A×B, where either A has Lebesgue measure 0 or B is a single point. (wikipedia.org)

###### called a Radon measure

- The measure m is called a Radon measure if it is inner regular and locally finite. (wikipedia.org)

###### Dirac

- For example, the Dirac delta function is a singular measure. (wikipedia.org)
- An important special case is where μ is a probability measure or, even more specifically, the Dirac delta function. (wikipedia.org)
- however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a delta sequence. (wikipedia.org)
- Let δx denote the Dirac measure centred on some fixed point x in some measurable space (X, Σ). (wikipedia.org)
- The Dirac measure is not sigma-finite A discrete measure is similar to the Dirac measure, except that it is concentrated at countably many points instead of a single point. (wikipedia.org)

###### Radon

- These conditions guarantee that the measure is "compatible" with the topology of the space, and most measures used in mathematical analysis and in number theory are indeed Radon measures. (wikipedia.org)
- Another approach to measure theory is to restrict to locally compact Hausdorff spaces, and only consider the measures that correspond to positive linear functionals on the space of continuous functions with compact support (some authors use this as the definition of a Radon measure). (wikipedia.org)
- If there is no restriction to non-negative measures and complex measures are allowed, then Radon measures can be defined as the continuous dual space on the space of continuous functions with compact support. (wikipedia.org)
- If such a Radon measure is real then it can be decomposed into the difference of two positive measures. (wikipedia.org)
- Furthermore, an arbitrary Radon measure can be decomposed into four positive Radon measures, where the real and imaginary parts of the functional are each the differences of two positive Radon measures. (wikipedia.org)
- The theory of Radon measures has most of the good properties of the usual theory for locally compact spaces, but applies to all Hausdorff topological spaces. (wikipedia.org)
- The idea of the definition of a Radon measure is to find some properties that characterize the measures on locally compact spaces corresponding to positive functionals, and use these properties as the definition of a Radon measure on an arbitrary Hausdorff space. (wikipedia.org)
- It is possible to extend the theory of Radon measures to non-Hausdorff spaces, essentially by replacing the word "compact" by "closed compact" everywhere. (wikipedia.org)
- When the underlying measure space is a locally compact topological space, the definition of a Radon measure can be expressed in terms of continuous linear functionals on the space of continuous functions with compact support. (wikipedia.org)
- Hence, δx is also a Radon measure. (wikipedia.org)

###### Borel

- One way to do this is to define a measure on the Borel sets of the topological space. (wikipedia.org)
- If X is a Hausdorff topological space with its Borel σ-algebra, then δx satisfies the condition to be an inner regular measure, since singleton sets such as {x} are always compact. (wikipedia.org)
- The Borel measure on the Euclidean space Rn can be obtained as the product of n copies of the Borel measure on the real line R. Even if the two factors of the product space are complete measure spaces, the product space may not be. (wikipedia.org)

###### Hausdorff

- Hausdorff measures. (wikipedia.org)
- 0, where Hs denotes the s-dimensional Hausdorff measure. (wikipedia.org)

###### subsets

- Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set X (see Definition below). (wikipedia.org)
- It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. (wikipedia.org)
- 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. (wikipedia.org)

###### mathematical

- In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. (wikipedia.org)
- Empirical measures are relevant to mathematical statistics. (wikipedia.org)
- In mathematical analysis, a Young measure is a parameterized measure that is associated with certain subsequences of a given bounded sequence of measurable functions. (wikipedia.org)

###### theory

- Measure Theory, Fourth Edition. (wikipedia.org)
- In the theory of vector measures, Lyapunov's theorem states that the range of a (non-atomic) vector measure is closed and convex. (wikipedia.org)
- Measure Theory Volume 2. (wikipedia.org)
- Ergodic theory considers measures that are invariant under, or arise naturally from, a dynamical system. (wikipedia.org)
- In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables. (wikipedia.org)
- They are named after Laurence Chisholm Young who invented them, however, in terms of linear functionals already in 1937 still before the measure theory has been developed. (wikipedia.org)
- Indeed, one of the fundamental theorems in measure theory states that a pre-measure can be extended to a measure. (wikipedia.org)
- Measure Theory. (wikipedia.org)

###### topological spaces

- For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. (wikipedia.org)
- Young measures on topological spaces. (wikipedia.org)
- Conceptually, this is similar to defining the Cartesian product of sets and the product topology of two topological spaces, except that there can be many natural choices for the product measure. (wikipedia.org)

###### singular measure

- A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. (wikipedia.org)
- This article incorporates material from singular measure on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. (wikipedia.org)

###### topology

- A common problem is to find a good notion of a measure on a topological space that is compatible with the topology in some sense. (wikipedia.org)

###### hence

- Hence, fairness measures like Jain's fairness index cannot be applied, as the measurement scale requires to be a ratio scale with a clearly defined zero point (see examples of misuse for coefficients of variation). (wikipedia.org)

###### martingales

- The name "forward measure" comes from the fact that under the forward measure, forward prices are martingales, a fact first observed by Geman (1989) (who is responsible for formally defining the measure). (wikipedia.org)
- Compare with futures prices, which are martingales under the risk neutral measure. (wikipedia.org)
- Measures, Integral and Martingales. (wikipedia.org)

###### 2017

- Measure S, originally known as the Neighborhood Integrity Initiative, was considered by voters in the city of Los Angeles in the March 7, 2017, election. (wikipedia.org)

###### Integral

- TFP gaps are characterized as the integral of a strictly concave function with respect to an employment-weighted measure of distortions. (nber.org)
- The measure in quantum physics is the integration measure used for performing a path integral. (wikipedia.org)
- In the limit where the sum is becoming an integral, the concept of the measure described above is replaced by an integration measure. (wikipedia.org)

###### metrics

- Although no single definition of a similarity measure exists, usually such measures are in some sense the inverse of distance metrics: they take on large values for similar objects and either zero or a negative value for very dissimilar objects. (wikipedia.org)
- Fairness measures or metrics are used in network engineering to determine whether users or applications are receiving a fair share of system resources. (wikipedia.org)

###### outer measures

- Outer measures are not, in general, measures, since they may fail to be σ-additive. (wikipedia.org)

###### convex

- In fact, the range of a non-atomic vector measure is a zonoid (the closed and convex set that is the limit of a convergent sequence of zonotopes). (wikipedia.org)
- On 3 January 1922,Hiram A. Farrand received patent #1,402,589 for his concave-convex tape, a major improvement for spring pocket tape measures. (wikipedia.org)
- It was Farrand's concave-convex tape that went on to become the standard for the majority of pocket tape measure tapes today The first patented for a long tape measure in the United States was patent #29,096 issued July 10, 1860 to William H. Paine of Sheboygan, Wisconsin , and produced by George M. Eddy and Company of Brooklyn, New York. (wikipedia.org)

###### generalization

- In this sense, a measure is a generalization of the concepts of length, area, and volume. (wikipedia.org)

###### spaces

- Given two measure spaces, there is always a unique maximal product measure μmax on their product, with the property that if μmax(A) is finite for some measurable set A, then μmax(A) = μ(A) for any product measure μ. (wikipedia.org)

###### Sigma

- Thus, what is missing for a pre-measure to be a measure is that it is not necessarily defined on a sigma-algebra (or a sigma-ring). (wikipedia.org)

###### define

- In multiplying measures, some of which are infinite, we define the product to be zero if any factor is zero. (wikipedia.org)

###### fundamental

- A fundamental question underlying Measure S was not just its provisions, but the direction the city itself would take. (wikipedia.org)

###### subset

- In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. (wikipedia.org)

###### satisfies

- Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties: Non-negativity: For all E in Σ: μ(E) ≥ 0. (wikipedia.org)

###### countable

- The counting measure can be defined on any measurable set, but is mostly used on countable sets. (wikipedia.org)

###### example

- In general there are several problems with this: for example, such a measure may not have a well defined support. (wikipedia.org)
- This only works when h-measure is a child microformat, i.e. the example given actually makes no sense. (microformats.org)
- The empirical distribution function provides an example of empirical measures. (wikipedia.org)
- Here is an example where a product has more than one product measure. (wikipedia.org)

###### space

- Denied more lab space for her pioneering research, Nancy Hopkins whipped out her tape measure. (technologyreview.com)
- A triple (X, Σ, μ) is called a measure space. (wikipedia.org)
- A probability space is a measure space with a probability measure. (wikipedia.org)
- Cosine similarity is a commonly used similarity measure for real-valued vectors, used in (among other fields) information retrieval to score the similarity of documents in the vector space model. (wikipedia.org)

###### Tape

- A tape measure or measuring tape is a flexible ruler and used to measure distance. (wikipedia.org)
- Surveyors use tape measures in lengths of over 100 m (300+ ft). (wikipedia.org)
- There are two basic types of tape measures with cases, spring return pocket tape measures and long tape measures. (wikipedia.org)
- Spring return pocket tape measures will generally fit in a pocket. (wikipedia.org)
- Pocket tape measures have a tape one foot to fifteen feet in length and 1/4 to 3/4 inches across. (wikipedia.org)
- When most people think of a tape measure, they are thinking of the pocket tape measure. (wikipedia.org)
- A second tape measure design is what is called the long tape. (wikipedia.org)
- Surveyors also used 49 ½ feet of a 50-foot tape for 3 rods and 99 feet of a 100-foot tape measure for 6 rods. (wikipedia.org)
- Tape measures are often designed for specific uses or trades. (wikipedia.org)
- Tape measures that were intended for use in tailoring or dressmaking were made from flexible cloth or plastic. (wikipedia.org)
- These types of tape measures were mainly used for the measuring of the human's waist line. (wikipedia.org)
- This type of tape measure will have a floating tang or hook on the end to aid measuring. (wikipedia.org)
- A tape measure of 25 or even 100 feet can wind into a relatively small container. (wikipedia.org)
- The self-marking tape measure allows the user an accurate one hand measure. (wikipedia.org)
- See the Plumbing article (the "Equipment and Tools" category) to find out how tape measures are used in that profession. (wikipedia.org)
- The first record of people using a measuring device was by the Romans using marked strips of leather, but this was more like a regular ruler than a tape measure. (wikipedia.org)
- Bang's rule was the first attempt in the United States to make a spring return pocket tape measure. (wikipedia.org)
- It is there Farrand and William Wentworth Brown began mass-producing the tape measure. (wikipedia.org)
- It would only accurately measure a distance equal to the total length of the tape from beginning to end marker, a brass piece attached to the tape at a measured distance. (wikipedia.org)
- This feature was never patented, but Justus Roe and Sons produced tape measures, "Roe Electric Reel Tape Measures", with this feature until 1895 when they started etching or stamping increments and numbers on the tapes. (wikipedia.org)

###### passes

- In fact, if Measure O passes, the county would have little choice but to spend even more on what would likely be a long and costly legal battle. (latimes.com)
- If Measure O passes and this public money is delivered into private hands, voters and taxpayers will have no further say over how it is actually spent. (latimes.com)

###### given

- The opposite construction to the formation of the product of two measures is disintegration, which in some sense "splits" a given measure into a family of measures that can be integrated to give the original measure. (wikipedia.org)
- Sometimes there is also a unique minimal product measure μmin, given by μmin(S) = supA⊂S, μmax(A) finite μmax(A), where A and S are assumed to be measurable. (wikipedia.org)

###### produces

- Taking f(x)=1 for all x in X produces the counting measure. (wikipedia.org)

###### interval

- QoE may be measure on interval scales. (wikipedia.org)

###### respect

- In finance, a T-forward measure is a pricing measure absolutely continuous with respect to a risk-neutral measure but rather than using the money market as numeraire, it uses a bond with maturity T. The use of the forward measure was pioneered by Farshid Jamshidian (1987), and later used as a means of calculating the price of options on bonds. (wikipedia.org)

###### infinite

- In this case the measure may be finite or infinite. (wikipedia.org)

###### general

- The counting measure is a special case of a more general construct. (wikipedia.org)
- General integration and measure. (wikipedia.org)

###### precise

- The responses above answer your precise question - I agree that measuring the torque at the motor with a motor mount which allows for that is probably your best (budget) option. (physicsforums.com)

###### continuous

- A singular continuous measure. (wikipedia.org)

###### real

- Opponents, who included many advocacy groups for the homeless as well as the city's business community, building trades unions and developers, said that while the measure addressed some real problems, it went too far and would have not only prevented the construction of new affordable housing but made the city's overall quality of life worse by aggravating an existing housing shortage. (wikipedia.org)
- The sheriff's department also attempted to block a mailer in support of Measure S designed to look like an eviction notice after many recipients mistook it for a real one. (wikipedia.org)
- In statistics and related fields, a similarity measure or similarity function is a real-valued function that quantifies the similarity between two objects. (wikipedia.org)

###### provision

- gathered self-reported data on exposure of Australian workers to vibration, and data on the provision of control measures for vibration in the workplace. (safeworkaustralia.gov.au)

###### correspond

- Size related distortions are shown to correspond to a mean preserving spread of this measure, explaining the stronger effects on TFP found in the literature. (nber.org)

###### fact

- Indeed, the fact that we chose to measure in the first place carries causal weight. (edweek.org)
- In fact, according to an analysis by the county counsel's office, Measure O would not reimburse private hospitals for their charitable care to the poor but for bad debts left unpaid by any patient. (latimes.com)

###### integration

- This makes it possible to develop measure and integration in terms of functional analysis, an approach taken by Bourbaki (2004) and a number of other authors. (wikipedia.org)
- Introduction to measure and integration. (wikipedia.org)