In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Consider a set and a σ-algebra on Then the tuple is called a measurable space.
Note that in contrast to a measure space, no measure is needed for a measurable space.
Look at the set:
One possible -algebra would be:
Then is a measurable space. Another possible -algebra would be the power set on :
With this, a second measurable space on the set is given by
If is finite or countably infinite, the -algebra is most often the power set on so This leads to the measurable space
If is a topological space, the -algebra is most commonly the Borel -algebra so This leads to the measurable space that is common for all topological spaces such as the real numbers
The term Borel space is used for different types of measurable spaces.
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.
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral.
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair is called a measurable space. The σ-algebras are a subset of the set algebras; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition.
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.
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.
In this course, various aspects of probability theory are considered. The first part is devoted to the main theorems in the field (law of large numbers, central limit theorem, concentration inequaliti
Solar Decathlon Europe (SDE) is an international multidisciplinary competition in which 20 university teams build and operate energy-efficient solar-powered houses. The aim of SDE is not only scientific but also educational and divulgative, making visitors ...