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Mesure (mathématiques)

En mathématiques, une mesure positive (ou simplement mesure quand il n'y a pas de risque de confusion) est une fonction qui associe une grandeur numérique à certains sous-ensembles d'un ensemble donné. Il s'agit d'un important concept en analyse et en théorie des probabilités. Intuitivement, la mesure d'un ensemble ou sous-ensemble est similaire à la notion de taille, ou de cardinal pour les ensembles discrets. Dans ce sens, la mesure est une généralisation des concepts de longueur, aire ou volume dans des espaces de dimension 1, 2 ou 3 respectivement. L'étude des espaces munis de mesures est l'objet de la théorie de la mesure. Lorsqu'on dispose d'une mesure μ sur un espace mesurable , on dit que le triplet est un espace mesuré ; Pour S ensemble mesurable (c'est-à-dire pour ), la valeur μ(S) est appelée la mesure de S ; Lorsque μ(X) est fini, on parle de mesure finie ou mesure bornée ; Lorsque μ(X) = 1, on parle de mesure de probabilité. Le triplet est alors appelé un espace probabilisé. Voir pour ce cadre l'article axiomes des probabilités. Lorsqu'il existe un recouvrement dénombrable de X par des sous-ensembles de mesure finie, c'est-à-dire, plus formellement, lorsqu'il existe une suite d'éléments de la tribu, tous de mesure finie, avec on parle de mesure σ-finie. Quitte à remplacer chaque par on peut supposer que la suite de sous-ensembles figurant dans la définition est croissante pour l'inclusion. Un sous-ensemble S de X est dit négligeable lorsqu'il est inclus dans un T appartenant à la tribu et de mesure nulle. La mesure μ est dite complète lorsque tout ensemble négligeable appartient à la tribu . Fonction mesurable. Les propriétés suivantes s'obtiennent sans mal à partir des axiomes précédents : Additivité : Si E1 et E2 sont deux ensembles mesurables disjoints, μ(E1 ∪ E2) = μ(E1) + μ(E2). Monotonie : Si E1 et E2 sont deux ensembles mesurables tels que E1 est un sous-ensemble de E2, alors μ(E1) ≤ μ(E2). Continuité à gauche : Si E1, E2, E3, ...

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Exercices associés (26)

Le liévre et la tortue

A hare and a turtle race over a distance L. The turtle starts running at constant speed

v_{\rm t}

and the hare at constant speed

v_{\rm h}

. The hare is not taking this seriously, and so

v_{\rm h} < v_{\rm t}

. When the turtle reaches a bridge that stands at a distance L′u2004

Cavité sous terre

Underground cavities can be detected by gravimetry.

At the surface of the earth, compute the earth’s gravitational field g₀ without any cavity and the gravitational field g₁ over a spherical cavity of radius R_(C) where the centre is at a depth of du2004≥u2004R_(C)u2006.
Let us assume that we can measure the gravitational field with a precision of (g₀−u2006g₁)/g₀u2004=u200410⁻⁶ and that we would like to detect a cavity just below the ground (R_(C)u2004=u2004d). What is the smal

Tube irrégulier

An incompressible liquid circulates in a tube. At a certain location, the section of the tube strongly decreases (cf. figure). The continuity equation indicates us that :

\frac{\partial\rho}{\partial t} + \nabla\cdot \rho \vec{v} = 0~.

Show that this is equivalent to the simpler equation:

\quad \langle v_n\rangle=\frac{1}{S_n}\iint_{S_n}\vec{v}\cdot d\vec{\sigma} ~.$$ If we add to the figure the velocity vectors as well as the vect

Coquille sphérique

Calculate, in any point of space, the electrostatic potential

V(\bf{r})

and the field

\bf{E}(\bf{r})

created by a spherical shell uniformly charged with an internal radius a, external b, and of total charge u2005+u2005Q, centered at the origin O of the cartesian axes. Graphically represent the functions

V(\bf{r})

and

E_r(\bf{r})

(the radial component of

\bf{E}(\bf{r})

) Replace now the charged shell by an ideal conductor of similar shape. The conductor is neutral and isolated. A charge u2005+u2005

Boucler la boucle

A stuntwoman attempts to drive her car weighing mg around a vertical looping track of radius R. She enters the track at point P with velocity V⃗_(P) and we assume that she then free-wheels around the track without either braking or accelerating. There is no friction.

Write the car’s equations of motion for the entire loop.
Express the car’s mechanical energy at all points of the loop, and show that the mechanical energy is a simple integral of the motion.
Compute the reaction force from t

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Notebooks associés (10)

Simulating Electron Motion in 2D Materials Using the Drude Model - Basic

Drude model considers electrons small solid particles traveling through lattices as a stationary array of heavier and bigger atoms. The model use "mean free time" as a measure of the frequency an electron gets scattered. This jupyter notebook makes use of this concept and uses stochastic method to simulat electrons traveling through a 2D material of finite width but infinite length."

"#### First we setup the simulation parameters such as the external electric field, number of electrons (default

Simulating Electron Motion in 2D Materials Using the Drude Model - Advanced

Drude model considers electrons small solid particles traveling through lattices as a stationary array of heavier and bigger atoms. The model use "mean free time" as a measure of the frequency an electron gets scattered. This jupyter notebook makes use of this concept and uses stochastic method to simulate electrons traveling through a 2D material of finite width but infinite length."

First we setup the simulation parameters such as the external electric field, number of electrons (default to 1)

Sustainability Week 1: The Carbon Footprint of Algorithms – Solutions

{#Sustainability-Week-1:-the-Carbon-Footprint-of-Algorithms} How to use this notebook Simply read the text and follow the instructions.
This notebook contains code cells, which can be modified and must be executed to see the result of their content.
To execute a cell, select it and click on the play button (▶) in the tool bar, or type Shift + Enter or Ctr + Enter. As the variables contained in a cell are stored in memory, the order of execution of the cells is important! Notebook by M

Whistle-Activated Keylock System Implementation

Whistling Keylock[1]¶ In this notebook we will implement a simple "whistle-activated" keylock, that is, an access control system where the password is a pre-defined melody. To program the lock, you record your musical passphrase and then, to open the lock, you whistle it back. Although the implementation will be simple and skip most of the fine details that would be necessary to produce a truly robust system, we need to take into account at least the following points: * The syste

Fairness Week 2: COMPAS and Fairness in Machine Learning – Solutions

Fairness Week 2: Predicting recidivism¶{.anchor-link} {#Fairness-Week-2:-Predicting-recidivism}

How to use this notebook Simply read the text and follow the instructions.
This notebook contains code cells, which can be modified and must be executed to see the result of their content.
To execute a cell, select it and click on the play button (▶) in the tool bar, or type Shift + Enter or Ctr + Enter. As the variables contained in a cell are stor

Afficher plus

Exercices associés (26)

Le liévre et la tortue

A hare and a turtle race over a distance L. The turtle starts running at constant speed

v_{\rm t}

and the hare at constant speed

v_{\rm h}

. The hare is not taking this seriously, and so

v_{\rm h} < v_{\rm t}

. When the turtle reaches a bridge that stands at a distance L′u2004

Cavité sous terre

Underground cavities can be detected by gravimetry.

At the surface of the earth, compute the earth’s gravitational field g₀ without any cavity and the gravitational field g₁ over a spherical cavity of radius R_(C) where the centre is at a depth of du2004≥u2004R_(C)u2006.
Let us assume that we can measure the gravitational field with a precision of (g₀−u2006g₁)/g₀u2004=u200410⁻⁶ and that we would like to detect a cavity just below the ground (R_(C)u2004=u2004d). What is the smal

Tube irrégulier

An incompressible liquid circulates in a tube. At a certain location, the section of the tube strongly decreases (cf. figure). The continuity equation indicates us that :

\frac{\partial\rho}{\partial t} + \nabla\cdot \rho \vec{v} = 0~.

Show that this is equivalent to the simpler equation:

\quad \langle v_n\rangle=\frac{1}{S_n}\iint_{S_n}\vec{v}\cdot d\vec{\sigma} ~.$$ If we add to the figure the velocity vectors as well as the vect

Coquille sphérique

Calculate, in any point of space, the electrostatic potential

V(\bf{r})

and the field

\bf{E}(\bf{r})

V(\bf{r})

and

E_r(\bf{r})

(the radial component of

\bf{E}(\bf{r})

) Replace now the charged shell by an ideal conductor of similar shape. The conductor is neutral and isolated. A charge u2005+u2005

Boucler la boucle

Write the car’s equations of motion for the entire loop.
Express the car’s mechanical energy at all points of the loop, and show that the mechanical energy is a simple integral of the motion.
Compute the reaction force from t

Afficher plus

Mesure (mathématiques)

Chattez avec Graph Search

First we setup the simulation parameters such as the external electric field, number of electrons (default to 1)

Fairness Week 2: Predicting recidivism¶{.anchor-link} {#Fairness-Week-2:-Predicting-recidivism}

First we setup the simulation parameters such as the external electric field, number of electrons (default to 1)

Fairness Week 2: Predicting recidivism¶{.anchor-link} {#Fairness-Week-2:-Predicting-recidivism}

Adaptive vector-valued splines for the resolution of inverse problems

Uniform dimension results of level-sets of stable sheets

Particle methods for optimization over the space of measures