In mathematics, a field of sets is a mathematical structure consisting of a pair consisting of a set and a family of subsets of called an algebra over that contains the empty set as an element, and is closed under the operations of taking complements in finite unions, and finite intersections.
Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over " is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory.
Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.
A field of sets is a pair consisting of a set and a family of subsets of called an algebra over that has the following properties:
for all
as an element:
Assuming that (1) holds, this condition (2) is equivalent to:
Any/all of the following equivalent conditions hold:
for all
for all
for all integers and all
for all integers and all
In other words, forms a subalgebra of the power set Boolean algebra of (with the same identity element ).
Many authors refer to itself as a field of sets.
Elements of are called points while elements of are called complexes and are said to be the admissible sets of
A field of sets is called a σ−field of sets and the algebra is called a σ-algebra if the following additional condition (4) is satisfied:
Any/both of the following equivalent conditions hold:
for all
for all
For arbitrary set its power set (or, somewhat pedantically, the pair of this set and its power set) is a field of sets. If is finite (namely, -element), then is finite (namely, -element). It appears that every finite field of sets (it means, with finite, while may be infinite) admits a representation of the form with finite ; it means a function that establishes a one-to-one correspondence between and via : where and (that is, ).
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
The course is based on Durrett's text book
Probability: Theory and Examples.
It takes the measure theory approach to probability theory, wherein expectations are simply abstract integrals.
Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a
Learn to optimize on smooth, nonlinear spaces: Join us to build your foundations (starting at "what is a manifold?") and confidently implement your first algorithm (Riemannian gradient descent).
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values true and false, usually denoted 1 and 0, whereas in elementary algebra the values of the variables are numbers. Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬.
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic. Interior algebras form a variety of modal algebras. An interior algebra is an algebraic structure with the signature ⟨S, ·, +, ′, 0, 1, I⟩ where ⟨S, ·, +, ′, 0, 1⟩ is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called the interior of x.
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets and is . The symmetric difference of the sets A and B is commonly denoted by or The power set of any set becomes an abelian group under the operation of symmetric difference, with the empty set as the neutral element of the group and every element in this group being its own inverse.
We prove a sharp quantitative version of the Faber–Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit which measures by how much the STFT of a function fails to be optimally concentrated on an arbitrary set of pos ...
On ten loose handwritten folios dating back from April 1679, Leibniz gradually devised, in the course of three days, a full-blown theory of thought that nonetheless remained unpublished and still has received little attention from scholars. Conceiving of a ...
The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for log-correlated fields in 2D in the subcritical regime. By consid ...