Integrals of scalar functions against a vector measure and their applications to some questions of functional analysis and linear integral equations

We prove some assertions on the decomposition of indefinite integrals of scalar functions against a vector measure, as well as of continuous linear operators acting from a fundamental Banach space $X(T,\Sigma,\mu)$ to a Hilbert space $H$. Hence we deduce a representation theorem for continuous linear operators going from $X$ to $H$. These results are applied to most general linear integral equations of the form $\int\limits_Tx(t)d\nu=\varphi$, $x\in X$, $\varphi\in H$, $\nu\colon\Sigma\to H$, $\nu\ll\mu$. Such equations are equivalent to certain infinite systems of scalar integral equations and to infinite systems of linear algebraic equations.