Integration with respect to the canonical spectral measure in sequence spaces

Given a spectral measure $P$ acting in a locally convex space $X$, there is a subtle connection between the properties of $P$ and its associated space $\mathcal{L}^1(P)$ of $P$-integrable functions and of the topological properties of the underlying space $X$ and the space $L(X)$ of all continuous linear operators on $X$ (equipped with the strong operator topology). This paper makes a detailed study of the canonical spectral measure $P$ acting in a class of locally convex \textit{sequence spaces} $X\subseteq \mathbb{C}^\mathbb{N}$. Special emphasis is placed on developing criteria which guarantee the $\sigma$-additivity of $P$ and criteria which allow for an explicit identification of $\mathcal{L}^1(P)$. Moreover, certain desirable features of the integration map $f \mapsto\int fdP, f\in\mathcal{L}^1(P)$, are established which are not true for general spectral measures acting in arbitrary locally convex spaces $X$.