Improper Riemann integrals and uniformly regular matrices (Q1114794)

A row-finite matrix \((a_{n,k})\) is said to be uniformly regular iff (i) \(\lim_{n\to \infty}a_{n,k}=0\) uniformly in k; (ii) \(\lim_{n\to \infty}\sum_{k}a_{n,k}=1;\) (iii) \(\sup_{n=1}\sum_{k}| a_{n,k}| <\infty.\) A real function f on (0,1] is said to be dominantly integrable if this function is Riemann integrable on \([b,1]\) for any \(b\in (0,1)\) and the function defined by \(\hat f(t)=\sup \{| f(x)|:\quad t\leq x\leq 1\}\) is improperly Riemann integrable on \((0,1].\) The author gives some conditions under which \[ \lim_{n\to \infty}\sum_{k}a_{n,k}f(x_{n,k})=\int^{1}_{0}f(x)dx, \] where f is a dominantly integrable function on (0,1] and \((a_{n,k})\) is a row- finite uniformly regular matrix with \(a_{n,k}\geq 0.\)