The work " provides local constructions for characteristic classes based on higher dimensional relatives of the measurable Riemann mapping in dimension two and the Yang – Mills theory in dimension four.

The information mapping goal is measurable results that change the way people present information.

Let T be a linear mapping from the space of μ < sub > 1 </ sub >- integrable functions into the space of μ < sub > 2 </ sub >- measurable functions, and for 1 ≤ p, q ≤ ∞, define to be the operator norm of a continuous extension of T to

Every measurable entity maps into a cardinal function but not every cardinal function is the result of the mapping of a measurable entity.

** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

# If A is a Lebesgue measurable set, then it is " approximately open " and " approximately closed " in the sense of Lebesgue measure ( see the regularity theorem for Lebesgue measure ).

However, a measurable function is nearly a continuous function ; see Luzin's theorem.

Indeed, according to the Kochen-Specker theorem, in the quantum mechanical formalism it is impossible that, for a given quantum state, each one of these measurable properties ( observables ) has a determinate ( sharp ) value.

For Lebesgue measurable functions, the theorem can be stated in the following form:

In mathematics, the Radon – Nikodym theorem is a result in measure theory that states that, given a measurable space ( X, Σ ), if a σ-finite measure on ( X, Σ ) is absolutely continuous with respect to a σ-finite measure on ( X, Σ ), then there is a measurable function f on X and taking values in

* By the Gelfand-Neumark theorem, the category of localizable measurable spaces ( with measurable maps ) is equivalent to the category of commutative Von Neumann algebras ( with normal unital homomorphisms of *- algebras ).

Any set which has a well-defined Lebesgue measure is said to be " measurable ", but the construction of the Lebesgue measure ( for instance using Carathéodory's extension theorem ) does not make it obvious whether there exist non-measurable sets.

This is a kind of Riesz representation theorem which allows for a measure to be represented as a linear functional on measurable functions.

* Various theorems concerning convergence of families of measurable and holomorphic functions, such as Vitali convergence theorem

In measure theory, a branch of mathematics, the ham sandwich theorem, also called the Stone – Tukey theorem after Arthur H. Stone and John Tukey, states that given measurable " objects " in-dimensional space, it is possible to divide all of them in half ( with respect to their measure, i. e. volume ) with a single-dimensional hyperplane.

The Loomis-Sikorski theorem provides a Stone-type duality between abstract sigma algebras and measurable spaces.

In probability theory, the Helly – Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions.

In measure theory, an area of mathematics, Egorov's theorem establishes a condition for the uniform convergence of a pointwise convergent sequence of measurable functions.

The most common form of the theorem states that a measurable function on π is square integrable if and only if the corresponding Fourier series converges in the space L < sup > 2 </ sup >.

The first principle is based on the fact that the inner measure and outer measure are equal for measurable sets, the second is based on Lusin's theorem, and the third is based on Egorov's theorem.

In the mathematical field of real analysis, Lusin's theorem ( or Luzin's theorem, named for Nikolai Luzin ) states that every measurable function is a continuous function on nearly all its domain.

" Lebesgue shows that his conditions lead to the theory of measure and measurable functions and the analytical and geometrical definitions of the integral.

In 2011, The National Aquarium was honored with the Maryland Green Registry Leadership Award, as an organization that shows “ a strong commitment to sustainable practices, measurable results, and continuous improvement ” and was recognized by the Baltimore Business Journal and Smart CEO Magazine for exceptional green business practices in 2009..

* 1995 Public / Private Ventures Study research shows measurable, positive results on youth who have a Big Brother or Sister — seminal research in the field of youth mentoring

In 1970, Solovay constructed Solovay's model, which shows that it is consistent with standard set theory, excluding uncountable choice, that all subsets of the reals are measurable.

The " level of service " can also be specified as " target " and " minimum ," which allows customers to be informed what to expect ( the minimum ), while providing a measurable ( average ) target value that shows the level of organization performance.

: 1908 – Trouton-Rankine experiment shows that length contraction of an object according to one frame does not produce a measurable change of resistance in the object's rest frame

When applied to certain nuclear reactions, the equation shows that an extraordinarily large amount of energy will be released, much larger than in the combustion of chemical explosives, where the mass difference is hardly measurable at all.

A quantitative trait ( see quantitative trait locus ) is a measurable trait that shows continuous variation, such as height or weight.

1995-American Public / Private Ventures Study research shows measurable results on youth who have a Big Brother or Sister.

*: The condition number computed with this norm is generally larger than the condition number computed with square-summable sequences, but it can be evaluated more easily ( and this is often the only measurable condition number, when the problem to solve involves a non-linear algebra, for example when approximating irrational and transcendental functions or numbers with numerical methods.

For more than 40 years, Paul Ekman has supported the view that emotions are discrete, measurable, and physiologically distinct.

Second, unless the specimen is drawn into a fluoride tube or processed immediately to separate the serum or plasma from the cells, the measurable glucose will be gradually lowered by in vitro metabolism of the glucose at a rate of approximately 7 mg / dL / hr, or even more in the presence of leukocytosis.

This system provides a clear technique which can be employed by clinical and laboratory assessments to more accurately diagnose and prescribe therapy for lymphedema, as well as obtain measurable outcomes.

The peak power is not always readily measurable, however, and the measurement of the average power is more commonly performed by an instrument.

" As a practical matter, the stops display a measurable racial disparity: black and Hispanic people generally represent more than 85 percent of those stopped by the police, though their combined populations make up a small share of the city ’ s racial composition.

In mathematics, a complete measure ( or, more precisely, a complete measure space ) is a measure space in which every subset of every null set is measurable ( having measure zero ).

As part of the development of Lebesgue integration, Lebesgue invented the concept of measure, which extends the idea of length from intervals to a very large class of sets, called measurable sets ( so, more precisely, simple functions are functions that take a finite number of values, and each value is taken on a measurable set ).

However, recent research suggests the older, yellowing leaves have no more measurable salt content than the other, greener leaves.

There are normally 16. 5 days each year with measurable snow ( 0. 1 inch ( 2. 5 mm ) or more ).

However, measurable snowfalls every winter, and in many winters one or more intense storms called Nor ' easters may occasionally produce blizzard conditions with snowfalls of 1 – 2 feet ( 30 – 60 cm ) and near-hurricane force winds.

The term frequently implies a critique of the more extreme expressions of logical positivism and has been used by social scientists such as Friedrich Hayek, philosophers of science such as Karl Popper, and philosophers such as Hilary Putnam to describe the dogmatic endorsement of scientific methodology and the reduction of all knowledge to only that which is measurable.

Productivism is the belief that measurable economic productivity and growth is the purpose of human organization ( e. g., work ), and that " more production is necessarily good ".

His theory of " development as freedom " is one of several human development theories, that states that the growth of individual capital, that is, " talent ", " creativity " and " personal ingenuity ", is more significant than the growth of many other measurable quantities, e. g. production of products for commodity markets.

Reinforcement is a term in operant conditioning and behavior analysis for a process of strengthening a directly measurable dimension of behavior — such as rate ( e. g., pulling a lever more frequently ), duration ( e. g., pulling a lever for longer periods of time ), magnitude ( e. g., pulling a lever with greater force ), or latency ( e. g., pulling a lever more quickly following the onset of an environmental event )— as a function of the delivery of a stimulus ( e. g. money from a slot machine ) immediately or shortly after the occurrence of the behavior.

" Among the changes, OMB issued the instruction to " mark one or more races " after noting evidence of increasing numbers of interracial children and wanting to capture the diversity in a measurable way, and after having received requests by people who wanted to be able to acknowledge their or their children's full ancestry rather than identifying with only one group.

Behaviorism ’ s objection to introspection focused much more heavily on its lack of reliability and objectivity which conflicted with behaviorism ’ s focus on measurable behavior rather than consciousness or sensation.

When concentration of these compounds is too low to cause a measurable increase in turbidity when bound to antibody, more specialised methods must be used.

The reduction was more pronounced after three to four hours, and was measurable up to 24 hours after drinking the juice.

Men achieved their goal 22 % more often when they engaged in goal setting, ( a system where small measurable goals are being set ; such as, a pound a week, instead of saying " lose weight "), while women succeeded 10 % more when they made their goals public and got support from their friends.