Functional operators and families of set functions

Let $X$ be the $F$-space of the functions $x(t)$ defined on the measurable space $(T,\Sigma,\mu)$ with values in $B$-space $Y$. We consider the operators $f$ mapping $X$ to the $B$-space $Z$. $X$, $Y$, and $Z$ are considered over the scalar field $R$. To each operator $f$ is associated the family $\Phi_f$ of vector-valued functions $\Phi_X(e)\colon\Sigma\to Z$, $\Phi_X(e)=f(x\chi_e)$, $e\in\Sigma$. The characteristics of these families are given for various classes of operators. The relationship of convergence and continuation of the operators $f$ with convergence and continuation of the corresponding families $\Phi_f$ is considered. Riesz' theorem on integral representation of linear functionals is generalized.