Spectral integrals of Bernoulli generalized functionals

Let $\mathcal{S}\subset \mathcal{L}^2 \subset \mathcal{S}^*$ be the Gel'fand triple over the Bernoulli space, where elements of $\mathcal{S}^*$ are called Bernoulli generalized functionals. In this paper, we define integrals of Bernoulli generalized functionals with respect to a spectral measure (projection operator-valued measure) in the framework of $\mathcal{S}\subset \mathcal{L}^2 \subset \mathcal{S}^*$, and examine their fundamental properties. New notions are introduced, several results are obtained and examples are also shown.