In mathematics, and specifically in measure theory, equivalence is a notion of two measures being qualitatively similar. Specifically, the two measures agree on which events have measure zero. Let and be two measures on the measurable space and let and be the sets of -null sets and -null sets, respectively. Then the measure is said to be absolutely continuous in reference to if and only if This is denoted as The two measures are called equivalent if and only if and which is denoted as That is, two measures are equivalent if they satisfy Define the two measures on the real line as for all Borel sets Then and are equivalent, since all sets outside of have and measure zero, and a set inside is a -null set or a -null set exactly when it is a null set with respect to Lebesgue measure. Look at some measurable space and let be the counting measure, so where is the cardinality of the set a. So the counting measure has only one null set, which is the empty set. That is, So by the second definition, any other measure is equivalent to the counting measure if and only if it also has just the empty set as the only -null set.

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Cours associés (14)
MATH-432: Probability theory
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.
ENG-474: Impact studies
Présentation des bases des études d'impact, du contexte et des outils d'évaluation de chacun des sujets et des chapitres. Illustration par de nombreux cas réels, et par un travail de groupe. Discussio
COM-309: Introduction to quantum information processing
Information is processed in physical devices. In the quantum regime the concept of classical bit is replaced by the quantum bit. We introduce quantum principles, and then quantum communications, key d
Afficher plus
Séances de cours associées (38)
Enchevêtrement: Inégalités de Bell
Couvre l'enchevêtrement, les inégalités de Bell, les exigences de l'ESCS, le paradoxe de l'EPE et la vérification expérimentale en mécanique quantique.
Formulation variationnelle : Mesures d'information
Explore la formulation variationnelle pour mesurer le contenu de l'information et la divergence entre les distributions de probabilité.
Fondements de l'hydrologie
Introduit les principes fondamentaux de l'hydrologie, couvrant l'humidité du sol, la modélisation des précipitations et les systèmes de réservoirs.
Afficher plus
Publications associées (47)

The Impact of Data Persistence Bias on Social Media Studies

Tugrulcan Elmas

Social media studies often collect data retrospectively to analyze public opinion. Social media data may decay over time and such decay may prevent the collection of the complete dataset. As a result, the collected dataset may differ from the complete data ...
New York2023

Nothing New Under the Sun: Novelty Constructs and Measures in Social Studies

Davide Bavato

The concept of novelty is central to questions of creativity, innovation, and discovery. Despite the prominence in scientific inquiry and everyday discourse, there is a chronic ambiguity over its meaning and a surprising variety of empirical measures, whic ...
Emerald Publishing Limited2022

Evaluating objective measures of impairment to trunk strength and control for cross-country sit skiing

Benedikt Fasel

In Paralympic cross-country sit skiing, athlete classification is performed by an expert panel, so it may be affected by subjectivity. An evidence-based classification is required, in which objective measures of impairment must be identified. The purposes ...
SPRINGER LONDON LTD2021
Afficher plus

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.