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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In operator theory, Naimark''s dilation theorem is a result that characterizes positive operator valued measures. It can be viewed as a consequence of Stinespring''s dilation theorem.In the mathematical literature, one may also find other results that bear Naimark''s name. Let X be a compact Hausdorff space, H be a Hilbert space, and L(H) the Banach space of bounded operators on H. A mapping E from the Borel -algebra on X to L(H) is called a operator-valued measure if it is weakly countably additive, that is, for any disjoint sequence of Borel sets {Bi}, we have langle E (cup _i B_i) x, y rangle = sum_i langle E (B_i) x, y rangle, langle E (cup _i B_i) x, y rangle = sum_i langle E (B_i) x, y rangle