The aim of this paper is to prove the completeness of \(L_\textbf{m}^p\) space of intuitionistic fuzzy observables with corresponding intuitionistic fuzzy metric \(\rho _{IF}\) . We work in an intuitionistic fuzzy space \((\mathcal {F}, \textbf{m})\) with product, where \(\mathcal F\) is a family of intuitionistic fuzzy events and \(\textbf{m}\) is an intuitionistic fuzzy state. We use the Kolmogorov construction of probability space \((R^N,\sigma (\mathcal C), P)\) and the connection between the random variables and the intuitionistic fuzzy observables.