Chebyshev–Markov–Stieltjes inequalities

Remarks

For the upper bound one builds, by Hermite interpolation, a polynomial $q$ of degree $2m - 2$ with $q(\xi_j) = 1$ for $j \le k$, $q(\xi_j) = 0$ for $j > k$, and $q'(\xi_j) = 0$ for $j \ne k$. One shows that $q \ge 1$ on $(-\infty, \xi_k]$ and $q \ge 0$ on $\R$. Then $\nu((-\infty, \xi_k]) \le \int q \, d\nu = \int q \, d\mu = \sum_j \rho_j q(\xi_j) = \sum_{j \le k} \rho_j$, since the integrals of $q$ depend only on the moments; the lower bound follows by the reflection $x \mapsto -x$. The measure $\sum_j \rho_j \delta_{\xi_j}$ of Gaussian quadrature has the given moments, and it attains every bound. In particular, the mass that any such $\nu$ puts at the point $\xi_k$ is at most $\rho_k$. For example, for $m = 1$ with $m_0 = 1$ and $m_1 = 0$, one gets $\xi_1 = 0$ and $\rho_1 = 1$, and the inequalities say only $0 \le \nu((-\infty, 0)) \le \nu((-\infty, 0]) \le 1$. The positive definiteness of the Hankel matrix holds exactly when $\mu$ is supported on at least $m$ points; it is needed for $P_m$ and the numbers $\rho_j$ to exist.

Going deeper

For a student

The inequalities say that the distribution functions of all solutions of a truncated moment problem pass through a band around the staircase of the Gaussian quadrature measure. The bound on point masses, $\nu(\set{\xi_k}) \le \rho_k$, also follows from the extremal property of the Christoffel function.