Proposed proof of the Birkhoff–von Neumann theorem with a ne

[khalid92]

I have just found a very beautiful and short proof for the birkhoff-von Neuman theorem that gives a new probabilistic approach.

**Notations** :
Let,
be the set of permutations of the set ,
,
/ is a probability measure over be the convex hull of the set of permutation matrices.

A doubly stochastic matrix (also called bi-stochastic), is a square matrix of non negative real numbers, each of whose rows and columns sums to 1, wi will note their set .

**Birkhoff-von Neuman theorem** :
The Birkhoff–von Neumann theorem states that $D$ is the convex hull of the set of n$\times$n permutation matrices.

**Proof:**

We will start in the proof by the most hard direction : .
First, we will put the point on some properties of that will be useful as we go in the proof.

1. for any and .
2. for any and
.
3.

Now let ,
To solve the problem we will construct a probability measure that verifiy :

.

We will define our probability measure on any subset of as the following:

.
With .

It is clear that is positive and -additive.
Lets demonstrate that it verify the third axiom :



And we have :


=

So , and we are done.

Let's rappel that a permutation matrix can be written as :

, where

Now,







.Witch means that

Because,our proof of this direction is constructive, it suffices to do the back forward job to prove the other direction of the the problem.
To make it clear :
Let,



We have,

.

and .

**conclusion**

We demonstrated here, in addition to the Birkhof theorem, that any stochastic matrix can be associated to a class of probability measures on .And its coefficients can be interpreted as the probability off the events with respect to any of theses measures.
We can even go further and split , the set of probability measures over , into equivalence classes with and .With the equivalence relation .
We can then define an isomorphism that associate to every a unique .And try then to characterize entirely .