Construction of Multiple Stable Measures and Integrals Using Lepage Representation

Consider a symmetric α-stable, 0 < α < 2, random measure M with a Radon control measure, defined on (R, B). Let f be a Banach-valued deterministic function defined on Rn, symmetric in its arguments and vanishing on the diagonals. We provide a construction of the product measure $${M^{\left( n \right)}}\left( {d{x_1},...,d{x_n}} \right) = n!M\left({d{x_1}} \right)...M\left( {d{x_n}} \right)$$ and of the multiple stochastic integral $${I_n}\left( f \right) = \int_{{R^n}} f \left( {{x_1},...,{x_n}} \right)M\left( {d{x_1}}\right)...M\left( {d{x_n}} \right)$$ using a multiple Lepage representation.