Given a probability space, a filtration is a weakly increasing collection of sigma-algebras on,, indexed by some totally ordered set T, and bounded above by.

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

Occasionally, " almost all " is used in the sense of " almost everywhere " in measure theory, or in the closely related sense of " almost surely " in probability theory.

In algorithmic information theory ( a subfield of computer science ), the Kolmogorov complexity of an object, such as a piece of text, is a measure of the computational resources needed to specify the object.

In mathematics, specifically in measure theory, a Borel measure is defined as follows: let X be a locally compact Hausdorff space, and let be the smallest σ-algebra that contains the open sets of X ; this is known as the σ-algebra of Borel sets.

Category: Measures ( measure theory )

In Geometric measure theory such a smooth curve as the circle that can be approximated by small straight segments with a definite limit is termed a rectifiable curve.

This is totally spurious, since no matter who measured first the other will measure the opposite spin despite the fact that ( in theory ) the other has a 50 % ' probability ' ( 50: 50 chance ) of measuring the same spin, unless data about the first spin measurement has somehow passed faster than light ( of course TI gets around the light speed limit by having information travel backwards in time instead ).

Other major critiques are that the term is not defined well, and employs further terms that are not defined well, and therefore lacks explanatory power, that cultural imperialism is hard to measure, and that the theory of a legacy of colonialism is not always true.

The field of recursion theory, meanwhile, categorizes undecidable decision problems by Turing degree, which is a measure of the noncomputability inherent in any solution.

Null set was once a common synonym for " empty set ", but is now a technical term in measure theory.

Felix Hausdorff ( November 8, 1868 – January 26, 1942 ) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.

An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

Graph theory is also widely used in sociology as a way, for example, to measure actors ' prestige or to explore diffusion mechanisms, notably through the use of social network analysis software.

* Information theory and measure theory

In information theory, entropy is a measure of the uncertainty associated with a random variable.

In probability theory and statistics, kurtosis ( from the Greek word κυρτός, kyrtos or kurtos, meaning bulging ) is any measure of the " peakedness " of the probability distribution of a real-valued random variable.

* Lambda denotes the Lebesgue measure in mathematical set theory.

Perhaps, with my Uncle, she found a measure of economic security that she needed ; ;

But, here again, comparative benchmarks are lacking, and we do not know, in any case, what measure of profoundity and intensity to expect from healthy, young, secure and relatively inexperienced persons ; ;

The objective function is some measure of the increase in value of the stream by processing ; ;

The ultimate objective of American policy is to help establish a world in which there is the largest possible measure of freedom and justice and peace and material prosperity ; ;

Progressive measures taken during his kingship include: representatives of the commons, besides the nobility and clergy, were involved in governance ; the end of preventive arrests such that henceforward all arrests had to be first presented to a judge to determine the detention measure ; and fiscal innovation, such as negotiating extraordinary taxes with the mercantile classes and direct taxation of the Church, rather than debasement of the coinage.

As a unit of measure, an acre has no prescribed shape ; any perimeter enclosing 43, 560 square feet is an acre in size.

Digital ammeter designs use an analog to digital converter ( ADC ) to measure the voltage across the shunt resistor ; the digital display is calibrated to read the current through the shunt.

For m ≥ 4, however, it grows much more quickly ; even A ( 4, 2 ) is about 2, and the decimal expansion of A ( 4, 3 ) is very large by any typical measure.

Anemometers can be divided into two classes: those that measure the wind's speed, and those that measure the wind's pressure ; but as there is a close connection between the pressure and the speed, an anemometer designed for one will give information about both.

In baseball statistics, on-base percentage ( OBP ; sometimes referred to as on-base average / OBA, as the statistic is rarely presented as a true percentage ) is a measure of how often a batter reaches base for any reason other than a fielding error, fielder's choice, dropped / uncaught third strike, fielder's obstruction, or catcher's interference ( the latter two are ignored as either times-on-base ( TOB ) or plate appearances in calculating OBP ).

* Absence of common measure of value: In a monetary economy, money plays the role of a measure of value of all goods, so their values can be measured against each other ; this role may be absent in a barter economy.

In this case, is the smallest σ-algebra that contains the open intervals of R. While there are many Borel measures μ, the choice of Borel measure which assigns for every interval is sometimes called " the " Borel measure on R. In practice, even " the " Borel measure is not the most useful measure defined on the σ-algebra of Borel sets ; indeed, the Lebesgue measure is an extension of " the " Borel measure which possesses the crucial property that it is a complete measure ( unlike the Borel measure ).

`` Dearly beloved '', he preached, `` unless you repent of your sins in a measure, and become converted to a degree, you will, I regret to say, be damned to a more or less extent ''.

The ceaseless effort to understand and measure the distance mankind has traversed since its primitive anthropological status offers a more durable sort of drama.

With more advanced equipment, but still cheap in comparison to professional setups, amateur astronomers can measure the light spectrum emitted from astronomical objects, which can yield high-quality scientific data if the measurements are performed with due care.

Ara Vartanian, the chairman of the Armenian Trade and Industry Chamber, thinks that this measure is a far more objective indicator of the tourist influx into the country.

In the 16th century, Tycho Brahe used improved instruments, including large mural instruments, to measure star positions more accurately than previously, with a precision of 15 – 35 arcsec.

A more precise way to measure resistance is to place an object within an artificial, uniform stream of air where the velocity is known.

A more refined measure for this exercise is to weigh the completed bridge rather than measure against a fixed quantity of materials provided and determine the multiple of this weight that the bridge can carry, a test that emphasizes economy of materials and efficient glue joints ( see balsa wood bridge ).

In more general mathematical settings, the Boltzmann distribution is also known as the Gibbs measure.

The Anabaptist tradition was largely ostracized by the other Protestant parties at the time, but has achieved a measure of affirmation in more recent history.

To measure how well the program is exercised by a test suite, one or more coverage criteria are used.

Adult dyslexics can read with good comprehension, but they tend to read more slowly than non-dyslexics and perform more poorly at spelling and nonsense word reading, a measure of phonological awareness.

It is a local measure of its " outgoingness "— the extent to which there is more exiting an infinitesimal region of space than entering it.

A more comprehensive approach to measuring cost-effectiveness or efficacy of malarial control would not only measure the cost in dollars of the project, as well as the number of people saved, but would also consider ecological damage and negative aspects of insecticide use on human health.

* Equivalent variation, a measure of how much more money a consumer would pay before a price increase to avert the price increase

If the number of counts is not very large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time.

They emphasize that not every film noir embodies all five attributes in equal measure — one might be more dreamlike ; another, particularly brutal.

But in measuring a wavy fractal curve such as the one in Figure 2, one would never find a small enough straight segment to conform to the curve, because the wavy pattern would always re-appear, albeit at a smaller size, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve.

Then they voted a measure far more radical, declaring themselves the National Assembly, an assembly not of the Estates but of " the People.

The orchestra on the other hand is more repetitive and pulse like: the left hand continually plays the high A and it is not until the 5th measure where another note is added, but the A continues to be played throughout always on the off beat.

Other tools are used to prepare yarn for knitting, to measure and design knitted garments, or to make knitting easier or more comfortable.

For this measure, higher kurtosis means more of the variance is the result of infrequent extreme deviations, as opposed to frequent modestly sized deviations.

These higher-order averages are more difficult to measure, but can yield additional information about molecular ordering.