With the development of quantum mechanical applications and extension to non- The foundations of quantum mechanics were made by Werner Heisenberg , Max Planck and Erwin Shrodinger in the first half of the twentieth century . In 1999 Parthasarathy consider the positive operator valued measure extremal in the case of finite space which is defined as fallows , if $X$ be a locally compact set and consider $O(X)$ as a $\\sigma$-algebra of borel sets of $X$ , then the set function $\u$ from $O(X)$ to $B(H)$ is called a Positive Operator Valued Measure (POVM) on $X$ if (1)for every $E \\in O(X)$ , $\u(E) \\in Eff(H)$ (2) for every countable disjoint collection $\\lbrace E_k\\rbrace_{k\\in \\mathbb{N}} \\subseteq O(X)$ , $\u(\\cup_{k\\in \\mathbb{N}} E_k)=\\sum_{k\\in \\mathbb{N}} \u(E_k)$ and (3) $\u(X) =1 \\in B(H)$\\cite{26} . In 2007 Chiribella and Shelingemann proved that a probability measure is an extremal if and only if it is degenerated distribution and finally in 2011 Farenick , Plosker and Smith consider positive operator valued measure extremal and generalized the above theorem in the non- This thesis is an extension (and generalization) of the work(s) done by Farenick , Plosker and Smith(\\cite{26}) . In this regard positive operator valued measure with finite support as a dense subset of $POVM_{H}(X)$ are studied . The first main result is studied in this thesis is as follows; if $\u$ belong to $POVM_H(X)$ , then there is a unital completely positive linear map $\\phi_\u$ from $C(X)\\otimes B(H)$ to $B(H)$ such that for every $f$ belong to $C(X)\\otimes B(H)$ , $$\\phi_{\u}(f)=\\int_X fd\u . $$ On the other hand convex combination $\\sum_{i=1}^{m} \\lambda_i\u_i$ Where $\u_1 , \u_2 , ..., \u_m$ are belong to $POVM_H(X)$ is called a random positive operator valued measure If the coefficients of a convex combination $\\{\\lambda_1 , \\lambda_2 , ..., \\lambda_m\\}$ belong to a discrete probability distribution . The concept of the In non- In addition proved that positive operator valued measure with finite support are dense in $POVM_{H}(X)$ and the equivalences of extreme point are studied . Finally the We introduce some basic definitions and concepts and requirements such as positive operator valued measure and $C^*$-convex combination in detail and explain equivalence of sharp measure and $C^*$-extreme points of $POVM_H(X)$ which a measure $\u$ is called sharp if there exist pairwise orthogonal projections $q_{1},...,q_{n}$ in $B(H)$ and distinct $x_{1},...,x_{n}$ in $X$ such that $\u=\\sum_{j=1}^{n} \\delta_{x_j}q_j$ .