A general framework for dealing with numerical measurements in an approximate, uncertain, or fuzzy environment is presented. A fuzzy measurement is defined. It possesses several unique properties, which arise from its physical nature and distinguish it from concepts such as the fuzzy number. These properties, which include the fuzzy correlation term and the fuzzy Equality relation, follow directly from physical considerations. The introduction of the fuzzy correlation term provides a mathematical tool for representing any correlation relations, which may exist between different fuzzy measurements. The main function of the fuzzy correlation term is to eliminate, or filter out, measurement values that are unlikely, given other fuzzy measurements. Thus, using the fuzzy correlation term, the range of possible measurement values is limited by physical realities. The information represented by the fuzzy correlation term is shown to be of great value in providing a wider picture of reality than it is possible to obtain by simply considering individual fuzzy measurements. Arithmetic operations on fuzzy measurements and functions of fuzzy measurements are also discussed, leading to the derivation of the fuzzy Riemann integral and its applications.