The Modulus of Oscillation of a Function about Number Sequences and Its Applications

We consider a characteristic simultaneously reflecting certain properties of Riemann integrable functions $f$ on the closed interval $[0,1]$ and properties of some sequence $X=\{x_n\}$points on $[0,1]$. The properties of functions are expressed by characteristics similar to the modulus of continuity, mean oscillation modulus, and the modulus of variation, while the properties of sequences are characterized by notions of maximal deviation and deviation in $L_p$. This characteristic is used to estimate the error $R_N(f,X)$ of the quadrature formula
$$ \int_0^1 f(x)\,dx=\frac{1}{N} \sum_{n=1}^N f(x_n)-R_N(f,X) $$
and to formulate condition for the uniform distribution of number sequences and the Riemann integrability of functions. All of the obtained main estimates are extremal.