Tribe produced

Definition

Being given two measurable spaces

(\ Omega_1, \ mathcal {T} _1)

and

(\ Omega_2, \ mathcal {T} _2)

, the tribe produced, noted

\ mathcal {T} _1 \ otimes \ mathcal {T} _2

, makes it possible to give a structure of space measurable to space produced

\ Omega_1 \ times \ Omega_2

; it is in the following way defined:

$\ mathcal {T} _1 \ otimes \ mathcal {T} _2$ is the generated tribe by the measurable paving stones $R=R_1 \ times R_2$ where $R_1 \ in \ mathcal {T} _ 1, R _2 \ in \ mathcal {T} _2$ or, in an equivalent way, the smallest tribe containing the measurable paving stones
one can also define it as the smallest measurable tribe making the projections $pr_1$ and $pr_2$ defined by: $pr_i (\ omega_1, \ omega_2) = \ omega_i, \ i=1, 2$

It is shown very easily that an application F, definite on a measurable space $(\ Omega, \ mathcal {has})$ with value in space produced $(E_1 \ times E_2, \ mathcal {T} _1 \ otimes \ mathcal {T} _2)$ measurable for the tribe is produced if and only if the coordinated applications $f_i$ are, each one, measurable for the tribes $\ mathcal {T} _i$ .

Example: tribe borélienne produced

Being given two topological spaces

(\ Omega_1, \ mathcal {O} _1)

and

(\ Omega_2, \ mathcal {O} _2)

provided with their respective tribes boréliennes

\ mathcal {B} _1

and

\ mathcal {B} _2

. There are then two ways natural to give to the product

\ Omega_1 \ times \ Omega_2

a structure of space measurable:

starting from the tribe produced $\ mathcal {B} _1 \ otimes \ mathcal {B} _2$
starting from the tribe borélienne generated by the topological structure $\ mathcal {O} _1 \ otimes \ mathcal {O} _2$ , noted $\ mathcal {B} (\ mathcal {O} _1 \ otimes \ mathcal {O} _2)$ .

One always has: $\ mathcal {B} _1 \ otimes \ mathcal {B} _2 \ subseteq \ mathcal {B} (\ mathcal {O} _1 \ otimes \ mathcal {O} _2)$ .

Indeed, projections

pr_i

continuous for topology are produced, therefore measurable for the tribe borélienne; the tribe produced being the smallest tribe making measurable projections one obtains desired inclusion.

If topological spaces $(\ Omega_i, \ mathcal {O} _i)$ are at countable base then $\ mathcal {B} _1 \ otimes \ mathcal {B} _2 = \ mathcal {B} (\ mathcal {O} _1 \ otimes \ mathcal {O} _2)$ .

Indeed, is U open of

\ mathcal {O} _1 \ otimes \ mathcal {O} _2

, then U is a countable union of measurable paving stones of the form

U_ {1} \ times U_ {2}

(because they form a countable base of topology produced): consequently

U \ in \ mathcal {B} _1 \ otimes \ mathcal {B} _2,

from where

\ mathcal {B} (\ mathcal {O} _1 \ otimes \ mathcal {O} _2) \ subseteq \ mathcal {B} _1 \ otimes \ mathcal {B} _2

Product of N tribes

The product of a finished number, let us say N, of tribes is defined in a similar way: it is about the smallest tribe containing the measurable paving stones

R_1 \ times… \ times R_n

. The properties stated for the product of 2 tribes extend without difficulty with the case from N tribes.

Countable product of tribes

If a measurable overall countable product now is considered, for example

\ mathbb {R} ^ {\ mathbb {NR}}

, the tribe produced

\ displaystyle {\ bigotimes_ {N \ in \ mathbb {NR}} \ mathcal {T} _n}

, defined on the product of the measurable units

\ displaystyle {\ prod_ {N \ in \ mathbb {NR}} (\ Omega_n, \ mathcal {T} _n})

is the tribe generated by the whole of the form

\ displaystyle {\ prod_ {N \ in \ mathbb {NR}} R_n}

where

R_n= \ safe Omega_n

for a finished number of indices N.

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