A probabilistic normed space (briefly PN space) has the conditions of a real normed space, that the norm of each element has the probabilistic interpretation in the set $\\Delta$ instead of $\\bbR$. such that $\\Delta$ is the set of all extended distribution functions $F:\\bbR\o\\bbA$ that are non-decreasing and left continuous. The ideas was introduced by a mathematician in 1963, whose name was \\v Serstnev. The importance of this replacement is, that $\\Delta$ should be a suitable extension of $\\bbRc$ (instead of $\\bbR$, for some reason) on topological, addition and scalar multiplication operation and order properties, and has sufficient condition for PN spaces definition. For this purpose $\\{\\vep_r\\mid r\\in\\bbRc\\}\\subset \\Delta$ is good replacement for $\\bbRc$ in $\\Delta$ that serves topological, addition and scalar multiplication operation and order properties of as a replacement of $\\bbR$ in $\\Delta$. In other words $\\vep$ is the replacement function from $\\bbRc$ into $\\Delta$. Fundamentally, for any real concept extension to probabilistic mode, this replacement is done. \\\\ In this paper, first introduce the notation of probabilistic valued measure and integrals, then $L^p$ spaces constructed, for $\\vep_1 \\leq p \\leq \\vep_\\infty$, and show that there exists natural probabilistic norm on these spaces, which turn them into complete probabilistic normed spaces.