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

Our starting point is the proof of the following property of stochastic matrices. Let $T=\{T_{i,j}\}$ be a $n\times m$ stochastic matrix such that for every couple of indexes $(i,j)$, there exists an index $l$ such that $T_{i,l}\neq T_{j,l}$. Then, there exists a real vector ${\mathbf k}=(k_1,k_2,\dots,k_m)$, $k_i\neq k_j$, $i\neq j$; $0<k_i\leq 1$, such that, $(T\,{\mathbf k})_i\neq (T\,{\mathbf k})_j$ if\, $i\neq j$. Then, we apply that property of stochastic matrices to probability theory. Let us consider an infinite sequence of linear functionals $\{T_i\}_{i\in\mathbb{N}}$, $T_if=\int f(t)\,d\mu_t(i)$, corresponding to an infinite sequence of probability measures $\{\mu_{(\cdot)}(i)\}_{i\in\mathbb{N}}$, on the Borel $\sigma$-algebra $\mathcal{B}([0,1])$ such that, $\mu_{(\cdot)}(i)\neq\mu_{(\cdot)}(j)$, $i,j\in\mathbb{N},\,\,i\neq j$. The property of stochastic matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function $f$ such that $$T_if=\int f(t)\,d\mu_t(i)\neq \int f(t)\,d\mu_t(j)=T_jf, \quad i,j\in\mathbb{N},\,i\neq j.$$ \noindent The relevance to quantum mechanics is pointed out.