Stochastic matrices and a property of the infinite sequences of linear functionals

Abstract

Our starting point is the proof of the following property of a particular class of matrices. Let T={Ti,j} be a n×m non-negative matrix such that ∑jTi,j=1 for each i. Suppose that for every pair of indices (i,j), there exists an index l such that Ti,l≠Tj,l. Then, there exists a real vector k=(k1,k2,…,km)T,ki≠kj,i≠j;0<ki⩽1, such that, (Tk)i≠(Tk)j ifi≠j.Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals {Ti}i∈N,Tif=∫f(t)dμt(i), corresponding to an infinite sequence of probability measures {μ(·)(i)}i∈N, on the Borel σ-algebra B([0,1]) such that, μ(·)(i)≠μ(·)(j),i,j∈N,i≠j. The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such thatTif=∫f(t)dμt(i)≠∫f(t)dμt(j)=Tjf,i,j∈N,i≠j.The relevance to quantum mechanics is showed.