Product measure

A product measure is a special measure on the product of measure spaces in mathematics. It is characterized in that it assigns a Cartesian product of sets, the product of the mass of the individual quantities. So is the -dimensional Lebesgue measure on the Borel - just the - fold product of the one-dimensional Lebesgue - Borel measure. In probability theory products of probability measures are used to model stochastic independence.

Construction of the product measure

Introduction

When thinking of the usual real line (ie, the - and -axis) thinks with the one-dimensional Lebesgue measure, so it is natural to define a measure on the plane such that the following holds for measurable quantities

Then follows in particular for the two-dimensional measure of a rectangle

The formula, that is the well-known formula, after which the area of ​​a rectangle is equal to the product of its side lengths.

Since even simple geometric figures such as triangles or circles, can not be represented as Cartesian products, the set function must be continued to a measure on a σ - algebra.

Products of two dimensions

For any two measurable spaces and is the first product - to define algebra. This is the by product of and

Generated - algebra, which is the smallest - algebra containing. (This step is necessary because the product itself is not an algebra in general, but only a half-ring. )

Be now and two measure spaces. One would then define analogous to the above example on the product σ - algebra is a measure which satisfies for all. A level which meets this condition will be called product measurement. In order to show that a measure is, for example, it can be represented as the integral of:

Such a measure always exists, as can be shown about the Maßerweiterungssatz of Carathéodory.

However, such a measure is not notwendigermaßen uniquely determined. However, if there are two σ - finite measure spaces, then also σ -finite and there exists a unique product measure. It is designated.

Products of finitely many dimensions

Be with and a family of measure spaces. A on the accompanying product - defined measure algebra is then called product measure of if for all

Applies. The existence of one shows by induction on using the product of two dimensions. Analogously, we obtain the uniqueness of continuation after the set if for all infinite.

Accordingly, we define the Produktmaßraum of.

Comments

• Using this definition, the principle of Cavalieri in its most general form can be formulated in which for each (almost everywhere) Lebesgue measurable subset.
• The sets of Fubini and Tonelli (ie not necessarily just for the Euclidean space ) are valid under the condition σ - finite measure spaces in general for measurable functions.
• For the uniqueness statement of is really necessary that both measure spaces - finite. If we set ( on [0,1] restricted Borel σ - algebra) and selected for the Lebesgue measure, for the non- σ - finite counting measure, so there are at least three different product measures, although still one of the is -finite measure spaces.
• The product measure of two complete measurements is in general not again completely, for example, is a -null set for each subset, but only for this amount lies in, ie it is
• In contrast, applies to the Borel σ - algebra for all.
• And spaces are two probability, each describing a random experiment, the product models the common experiment is to carry out the two individual experiments independently.