sigma-algebra

The discussion that follows parallels the most common expository approach to the Lebesgue integral. In this approach the theory of integration has two distinct parts: A theory of measurable sets and measures on these sets. A theory of measurable functions and integrals on these functions. Lebesgue integration - Measure theory. Measure theory initially was created to provide a detailed analysis of the notion of length of subsets of the real line and more generally area and volum ...

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Lebesgue integration, Lebesgue integration - Introduction, Lebesgue integration - Construction of the Lebesgue integral, Lebesgue integration - Measure theory, Lebesgue integration - Integration, Lebesgue integration - Intuitive interpretation, Lebesgue integration - Example, Lebesgue integration - Limitations of the Riemann integral, Lebesgue integration - Basic theorems of the Lebesgue integral, Lebesgue integration - Proof techniques, Lebesgue integration - Alternative formulations, Lebesgue integration - Quote

Read more here: » Lebesgue integration: Encyclopedia II - Lebesgue integration - Construction of the Lebesgue integral

In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph (). The cardinality of the natural numbers is aleph-null () (also aleph-naught, aleph-nought); the next larger cardinality is aleph-one , then and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number α, as will be described below. The concept goes back to Georg Cantor, who defined the notion of cardinality and realized t ...

Including:

• Aleph number - Aleph-null
• Aleph number - Aleph-one
• Aleph number - The continuum hypothesis
• Aleph number - Aleph-ω
• Aleph number - Aleph-α for general α

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Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as groups, rings and fields. Along with geometry and analysis, it is one of the three main branches of mathematics. Algebra - History. The origins of algebra can be trac ...

Including:

• Algebra - History
• Algebra - Classification
• Algebra - Algebras

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If f is non-negative, then ∫f dμ is precisely the area under the curve as measured by the product measure μ × λ where λ is the Lebesgue measure for R. One can try to circumvent measure theory entirely. The Riemann integral exists for any continuous function f of compact support. Then we use functional analysis to obtain the integral for more general functions. Let Cc be the space of all real-valued compactly supported continuous functions of RSee also:

Lebesgue integration, Lebesgue integration - Introduction, Lebesgue integration - Construction of the Lebesgue integral, Lebesgue integration - Measure theory, Lebesgue integration - Integration, Lebesgue integration - Intuitive interpretation, Lebesgue integration - Example, Lebesgue integration - Limitations of the Riemann integral, Lebesgue integration - Basic theorems of the Lebesgue integral, Lebesgue integration - Proof techniques, Lebesgue integration - Alternative formulations, Lebesgue integration - Quote

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To illustrate some of the proof techniques used in Lebesgue integration theory, we sketch a proof of the above mentioned Lebesgue monotone convergence theorem: Let {fk}k ∈ N be a non-decreasing sequence of non-negative measurable functions and put By the monotonicity property of the integral, it is immediate that: We now prove the inequa ...

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Lebesgue integration, Lebesgue integration - Introduction, Lebesgue integration - Construction of the Lebesgue integral, Lebesgue integration - Measure theory, Lebesgue integration - Integration, Lebesgue integration - Intuitive interpretation, Lebesgue integration - Example, Lebesgue integration - Limitations of the Riemann integral, Lebesgue integration - Basic theorems of the Lebesgue integral, Lebesgue integration - Proof techniques, Lebesgue integration - Alternative formulations, Lebesgue integration - Quote

Read more here: » Lebesgue integration: Encyclopedia II - Lebesgue integration - Proof techniques

Here we discuss the limitations of the Riemann integral and the greater scope offered by the Lebesgue integral. We presume a working understanding of the Riemann integral. With the advent of Fourier series, many analytical problems involving integrals came up whose satisfactory solution required exchanging infinite summations of functions and integral signs. However, the conditions under which the integrals are equal proved quite elusive in the Riemann framework. There are some other technical difficulties with the Riemann ...

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Lebesgue integration, Lebesgue integration - Introduction, Lebesgue integration - Construction of the Lebesgue integral, Lebesgue integration - Measure theory, Lebesgue integration - Integration, Lebesgue integration - Intuitive interpretation, Lebesgue integration - Example, Lebesgue integration - Limitations of the Riemann integral, Lebesgue integration - Basic theorems of the Lebesgue integral, Lebesgue integration - Proof techniques, Lebesgue integration - Alternative formulations, Lebesgue integration - Quote

Read more here: » Lebesgue integration: Encyclopedia II - Lebesgue integration - Limitations of the Riemann integral

The Lebesgue integral does not distinguish between functions which only differ on a set of μ-measure zero. To make this precise, functions f, g are said to be equal almost everywhere (or equal a.e.) iff If f, g are non-negative functions (possibly assuming the value +∞) such that f = g almost everywhere, then If f, g are functions such that f = g almost everywhere, then f is integrable iff gSee also:

Lebesgue integration, Lebesgue integration - Introduction, Lebesgue integration - Construction of the Lebesgue integral, Lebesgue integration - Measure theory, Lebesgue integration - Integration, Lebesgue integration - Intuitive interpretation, Lebesgue integration - Example, Lebesgue integration - Limitations of the Riemann integral, Lebesgue integration - Basic theorems of the Lebesgue integral, Lebesgue integration - Proof techniques, Lebesgue integration - Alternative formulations, Lebesgue integration - Quote

Read more here: » Lebesgue integration: Encyclopedia II - Lebesgue integration - Basic theorems of the Lebesgue integral

The integral of a function f between limits a and b can be interpreted as the area under the graph of f. This is easy to understand for familiar functions such as polynomials, but what does it mean for more exotic functions? In general, what is the class of functions for which "area under the curve" makes sense? The answer to this question has great theoretical and practical importance. As part of a general movement toward rigour in mathematics in the nineteenth century, attempts were made to put the integr ...

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Lebesgue integration, Lebesgue integration - Introduction, Lebesgue integration - Construction of the Lebesgue integral, Lebesgue integration - Measure theory, Lebesgue integration - Integration, Lebesgue integration - Intuitive interpretation, Lebesgue integration - Example, Lebesgue integration - Limitations of the Riemann integral, Lebesgue integration - Basic theorems of the Lebesgue integral, Lebesgue integration - Proof techniques, Lebesgue integration - Alternative formulations, Lebesgue integration - Quote

Read more here: » Lebesgue integration: Encyclopedia II - Lebesgue integration - Introduction

is the cardinality of the set of all countably infinite ordinal numbers, called ω1 or Ω. This definition implies (already in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set Ω (the standard example of a set o ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α, Aleph number - Fixed points of aleph, Aleph number - Popular culture

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Let X be a measurable space, let μ be a measure on X, and let N be a measurable set in X. If μ is a positive measure, then N is null if its measure μ(N) is zero. If μ is not a positive measure, then N is μ-null if N is |μ|-null, where |μ| is the total variation of μ; equivalently, if every measurable subset A of N satisfies μ(A)=0. For positives measures, this is equivalent to the definition given above; but for signed measures, this is stronger th ...

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Null set, Null set - Definition, Null set - Properties, Null set - In Lebesgue measure, Null set - Uses

Read more here: » Null set: Encyclopedia II - Null set - Definition

The qualitative properties of dynamical systems do not change under smooth change of coordinates (this is sometimes taken as a definition of qualitative): a singular point of the vector field (a point where v(x) = 0) will remain a singular point under smooth transformations; a periodic orbit is a loop in phase space and smooth deformations of the phase space cannot alter it being a loop. It is in the neighborhood of singular points and periodic orbits that the structure of a phase space of a dynamical system can be well ...

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Dynamical system, Dynamical system - Overview, Dynamical system - Basic definitions, Dynamical system - Linear dynamical systems, Dynamical system - Flows, Dynamical system - Maps, Dynamical system - Local dynamics, Dynamical system - Rectification, Dynamical system - Near periodic orbits, Dynamical system - Conjugation results, Dynamical system - Bifurcations, Dynamical system - Ergodic systems, Dynamical system - Chaos theory, Dynamical system - Formal definition, Dynamical system - Geometrical definition, Dynamical system - Measure theoretical definition, Dynamical system - Examples of dynamical systems

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In mathematics, a real-valued function f of a real variable is absolutely continuous if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint intervals [xk, yk], k = 1, ..., n satisfies then Every absolutely continuous function is uniformly continuous and, therefore, continuous. Every Lipschitz-continuous function is absolutely continuous. < ...

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Absolute continuity, Absolute continuity - Absolute continuity of real functions, Absolute continuity - Absolute continuity of measures, Absolute continuity - The connection between absolute continuity of real functions and absolute continuity of measures

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The following three axioms are known as the Kolmogorov axioms, after Andrey Kolmogorov who developed them. We have an underlying set Ω, a sigma-algebra F of subsets of Ω, and a function P assigning real numbers to members of F. The members of F are those subsets of Ω that are called "events". Probability axioms - First axiom. For any set i.e., for any event, That is, the probability of an event is a non-negative real number. Probab ...

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Probability axioms, Probability axioms - Kolmogorov axioms, Probability axioms - First axiom, Probability axioms - Second axiom, Probability axioms - Third axiom, Probability axioms - Lemmas in probability

Read more here: » Probability axioms: Encyclopedia II - Probability axioms - Kolmogorov axioms

A stochastic process is a random function, that is a random variable X defined on a probability space (Ω , Pr) with values in a space of functions F. The space F in turn consists of functions I → D. Thus a stochastic process can also be regarded as an indexed collection of random variables {Xi}, where the index i ranges through an index set I, defined on the probability space (Ω, Pr) and taking values on the same codomain D (often the real numbers R). This view of a stochastic process as an indexed colle ...

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Stochastic process, Stochastic process - Definition, Stochastic process - Examples, Stochastic process - Interesting special cases, Stochastic process - Constructing stochastic processes, Stochastic process - The Kolmogorov extension, Stochastic process - Separability or what the Kolmogorov extension does not provide

Read more here: » Stochastic process: Encyclopedia II - Stochastic process - Definition

is the cardinality of the set of all countably infinite ordinal numbers, called ω1 or Ω. This definition implies (already in ZF, Zermelo-Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set Ω (the standard example of a set o ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-one

An equivalent definition can be given in the language of measure-preserving dynamical systems. Let be a dynamical system, with T being the time-evolution or shift operator. Then, if for all , if one has then the system is called strong mixing. For shifts parameterized by a continuous variable instead of a discrete integer n, the same definition applies, with T − n replaced by Tg with g being the continuous-time parameter. A dynamic ...

See also:

Mixing mathematics, Mixing mathematics - Mixing in stochastic processes, Mixing mathematics - Mixing in dynamical systems, Mixing mathematics - Topological mixing, Mixing mathematics - Generalizations

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The following three axioms are known as the Kolmogorov axioms, after Andrey Kolmogorov who developed them. We have an underlying set Ω, a sigma-algebra F of subsets of Ω, and a function P assigning real numbers to members of F. The members of F are those subsets of Ω that are called "events". Probability axioms - First axiom. For any set i.e., for any event E, That is, the probability of an event is a non-negative real number. Probab ...

See also:

Probability axioms, Probability axioms - Kolmogorov axioms, Probability axioms - First axiom, Probability axioms - Second axiom, Probability axioms - Third axiom, Probability axioms - Lemmas in probability

Read more here: » Probability axioms: Encyclopedia II - Probability axioms - Kolmogorov axioms

Algebra may be roughly divided into the following categories: elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra); abstract algebra, sometimes also called modern algebra< ...

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Algebra, Algebra - History, Algebra - Classification, Algebra - Algebras

Read more here: » Algebra: Encyclopedia II - Algebra - Classification

In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it. For this purpose one traditionally uses a method called Kolmogorov extension. There is at least one alternative axiomatization of probability theory by means of expectations on C-star algebras of random variables. In this ...

See also:

Stochastic process, Stochastic process - Definition, Stochastic process - Examples, Stochastic process - Interesting special cases, Stochastic process - Constructing stochastic processes, Stochastic process - The Kolmogorov extension, Stochastic process - Separability or what the Kolmogorov extension does not provide

Read more here: » Stochastic process: Encyclopedia II - Stochastic process - Constructing stochastic processes

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number is the smallest upper bound of Aleph-ω is the first uncountable cardinal number that can be demonstrated within Zermelo-Fraenkel set theory not to be equal to the cardinality of the set of all real numbers; for any positive integer n we can consistently assume that , and moreover it is possible to assume is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality , meaning th ...

See also:

Aleph number, Aleph number - Aleph-null, Aleph number - Aleph-one, Aleph number - The continuum hypothesis, Aleph number - Aleph-ω, Aleph number - Aleph-α for general α, Aleph number - Fixed points of aleph, Aleph number - Popular culture

Read more here: » Aleph number: Encyclopedia II - Aleph number - Aleph-ω